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. t 4 15. 17. t 6 t 7 4 14. t 3 16. 18. t 4 t 4 3 39. 41. sin t 1 3 sint (a) csct (b) cost 1 5 cos t (a) sect (b) sin t 4 5 sin t (a) sint (b) 40. 42. csc t (b) cos t 3 4 cost (a) sect (b) cos t 4 5 cos t (a) (b) cost 333202_0402.qxd 12/7/05 11:02 AM Page 300 300 Chapter 4 Trigonometry In Exercises 43–52, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) sin 4 csc 1.3 cos1.7 csc 0.8 43. 45. 47. 49. 51. sec 22.8 tan 3 cot 1 cos2.5 sec 1.8 sin0.9 44. 46. 48. 50. 52. Estimation In Exercises 53 and 54, use the figure and a straightedge to approximate the value of each trigonometric function. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 53. (a) sin 5 (b) cos 2 54. (a) sin 0.75 (b) cos 2.5 Model It (co n t i n u e d ) (a) Complete the tableb) Use the table feature of a graphing utility to approximate the time when the weight reaches equilibrium. (c) What appears to happen to the displacement as t increases? 58. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring is given by yt 1 is the displacement (in feet) and is the time (in seconds). Find the displacement when (a) t 1 t 0, 2. 4 cos 6t, t 1 4, and (c) where (b) y t Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 59 and 60, determine whether 59. Because sint sin t, negative angle is a negative number. tan a tana 6 it can be said that the sine of a 1.75 1.50 1.25 2.00 1.00 0.75 0.50 0.25 6 |
.25 2.25 2.50 2.75 3.00 3.25 3.50 −0.8 −0.6 −0.4 3.75 4.00 4.25 0.8 0.6 0.4 0.2 −0.2 − 0.2 −0.4 −0.6 −0.8 0.2 0.4 0.6 0.8 1.2 60. 6.00 5.75 5.50 5.25 61. Exploration Let x1, y1 and t t1 circle corresponding to x2, y2 and respectively. x1, y1. x2, y2 (b) Make a conjecture about any relationship between (a) Identify the symmetry of the points and t t1, be points on the unit sin t1 and. sin t1 4.50 4.75 5.00 FIGURE FOR 53–56 cos t1 and. cos t1 (c) Make a conjecture about any relationship between Estimation In Exercises 55 and 56, use the figure and a straightedge to approximate the solution of each equation, where To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 0 ≤ t < 2. 55. (a) 56. (a) sin t 0.25 sin t 0.75 (b) (b) cos t 0.25 cos t 0.75 62. Use the unit circle to verify that the cosine and secant functions are even and that the sine, cosecant, tangent, and cotangent functions are odd. Skills Review In Exercises 63– 66, find the inverse function one-to-one function 3x 2 64. 63. f x 1 2 f. f 1 of the Model It 65. f x x2 4, x ≥ 2 66. 57. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is y t 1 given by is the displacement (in feet) and where is the time (in seconds). 4et cos 6t t y In Exercises 67–70, sketch the graph of the rational function by hand. Show all asymptotes. 67. 69. f x 2x x 3 f x x2 3x 10 2x2 8 68. 70. f x 5x x2 x 6 f x x3 6x2 |
x 1 2x2 5x 8 333202_0403.qxd 12/7/05 11:03 AM Page 301 4.3 Right Triangle Trigonometry Section 4.3 Right Triangle Trigonometry 301 What you should learn • Evaluate trigonometric functions of acute angles. • Use the fundamental trigonometric identities. • Use a calculator to evaluate trigonometric functions. • Use trigonometric functions to model and solve real-life problems. Why you should learn it Trigonometric functions are often used to analyze real-life situations. For instance, in Exercise 71 on page 311, you can use trigonometric functions to find the height of a helium-filled balloon. The Six Trigonometric Functions Our second look at the trigonometric functions is from a right triangle perspective. Consider a right triangle, with one acute angle labeled as shown in Figure 4.26. the three sides of the triangle are the hypotenuse, the Relative to the angle opposite side (the side opposite the angle ), and the adjacent side (the side adjacent to the angle ).,, Hypotenuse Side adjacent to θ FIGURE 4.26 Using the lengths of these three sides, you can form six ratios that define the six trigonometric functions of the acute angle. sine cosecant cosine secant tangent cotangent lies in the In the following definitions, it is important to see that first quadrant) and that for such angles the value of each trigonometric function is positive. 0 < < 90 Right Triangle Definitions of Trigonometric Functions Let be an acute angle of a right triangle. The six trigonometric functions of the angle are defined as follows. (Note that the functions in the second row are the reciprocals of the corresponding functions in the first row.) Joseph Sohm; Chromosohm sin opp hyp csc hyp opp cos adj hyp sec hyp adj tan opp adj cot adj opp The abbreviations opp, adj, and hyp represent the lengths of the three sides of a right triangle. opp the length of the side opposite adj the length of the side adjacent to hyp the length of the hypotenuse 333202_0403.qxd 12/7/05 11:03 AM Page 302 302 Chapter 4 Trigonometry Hypotenuse 4 θ 3 FIGURE 4.27 Example 1 Evaluating Trigonometric Functions Use the triangle in Figure 4.27 to find the values of the six trigonometric functions of. hyp2 opp |
2 adj2, it follows that Solution By the Pythagorean Theorem, hyp 42 32 25 5. So, the six trigonometric functions of are csc hyp opp sin opp hyp 4 5 5 4 cos adj hyp 3 5 tan opp adj 4 3 sec hyp adj 5 3 cot adj opp 3 4. Now try Exercise 3. Historical Note Georg Joachim Rhaeticus (1514–1576) was the leading Teutonic mathematical astronomer of the 16th century. He was the first to define the trigonometric functions as ratios of the sides of a right triangle. 45° 1 2 45° 1 FIGURE 4.28 In Example 1, you were given the lengths of two sides of the right triangle, but not the angle Often, you will be asked to find the trigonometric functions of a given acute angle To do this, construct a right triangle having as one of its angles... Example 2 Evaluating Trigonometric Functions of 45 Find the values of sin 45, cos 45, and tan 45. Solution Construct a right triangle having as one of its acute angles, as shown in Figure 4.28. Choose the length of the adjacent side to be 1. From geometry, you know that the other acute angle is also So, the triangle is isosceles and the length of the opposite side is also 1. Using the Pythagorean Theorem, you find the length of the hypotenuse to be 2. 45. 45 sin 45 opp hyp 1 2 cos 45 adj hyp 1 2 2 2 2 2 tan 45 opp adj 1 1 1 Now try Exercise 17. 333202_0403.qxd 12/7/05 11:03 AM Page 303 Section 4.3 Right Triangle Trigonometry 303 Example 3 Evaluating Trigonometric Functions of 30 and 60 and 30, 45, Because the angles 6, 4, and 3 60 occur frequently in trigonometry, you should learn to construct the triangles shown in Figures 4.28 and 4.29. Use the equilateral triangle shown in Figure 4.29 to find the values of cos 60, sin 30, cos 30. and sin 60, 30° 2 3 2 60° 1 1 FIGURE 4.29 Solution Use the Pythagorean Theorem and the equilateral triangle in Figure 4.29 to verify adj 1, the lengths of the sides shown in the figure. For opp 3, hyp 2. 60, you have and So, Te c h n o l o |
g y You can use a calculator to convert the answers in Example 3 to decimals. However, the radical form is the exact value and in most cases, the exact value is preferred. For 3 sin 60 opp 2 hyp adj 3, 30, sin 30 opp hyp 1 2 and opp 1, and cos 60 adj hyp 1 2. and hyp 2. So, cos 30 adj hyp 3 2. Now try Exercise 19. Sines, Cosines, and Tangents of Special Angles sin 30 sin sin 45 sin sin 60 sin 6 4 3 1 2 2 2 3 2 cos 30 cos cos 45 cos cos 60 cos 6 4 3 2 2 2 3 1 2 tan 30 tan tan 45 tan tan 60 tan 3 3 1 3 6 4 3 sin 30 1 2 In the box, note that 60 are complementary angles. In general, it can be shown from the right triangle definitions that cofunctions of complementary angles are equal. That is, if is an acute angle, the following relationships are true. This occurs because cos 60. and 30 sin90 cos tan90 cot sec90 csc cos90 sin cot90 tan csc90 sec 333202_0403.qxd 12/7/05 11:03 AM Page 304 304 Chapter 4 Trigonometry Trigonometric Identities In trigonometry, a great deal of time is spent studying relationships between trigonometric functions (identities). Fundamental Trigonometric Identities Reciprocal Identities sin 1 cos 1 csc sec tan 1 cot csc 1 sin sec 1 cos cot 1 tan Quotient Identities tan sin cos Pythagorean Identities sin2 cos2 1 cot cos sin 1 tan2 sec2 1 cot2 csc2 Note that sin2 represents sin 2, cos2 represents cos 2, and so on. Example 4 Applying Trigonometric Identities Let be an acute angle such that (b) using trigonometric identities. tan sin 0.6. Find the values of (a) cos and Solution a. To find the value of cos, use the Pythagorean identity sin2 cos2 1. So, you have 0.62 cos2 1 cos2 1 0.6 2 0.64 cos 0.64 0.8. Substitute 0.6 for sin. Subtract 0.62 from each side. Extract the positive square root. b. Now, knowing the sine and cosine of, you can find the tangent of to be tan sin cos |
0.6 0.8 0.75. 0.6 Use the definitions of cos and tan check these results., Now try Exercise 29. and the triangle shown in Figure 4.30, to 1 0.8 θ FIGURE 4.30 333202_0403.qxd 12/7/05 11:03 AM Page 305 Section 4.3 Right Triangle Trigonometry 305 Example 5 Applying Trigonometric Identities be an acute angle such that tan 3. Find the values of (a) cot and Let (b) sec using trigonometric identities. 10 3 θ 1 FIGURE 4.31 You can also use the reciprocal identities for sine, cosine, and tangent to evaluate the cosecant, secant, and cotangent functions with a calculator. For instance, you could use the following keystroke sequence to evaluate sec 28. 1 COS 28 ENTER The calculator should display 1.1325701. Solution a. cot 1 tan b. cot 1 3 sec2 1 tan2 sec2 1 32 sec2 10 sec 10 Reciprocal identity Pythagorean identity Use the definitions of check these results. cot and sec, and the triangle shown in Figure 4.31, to Now try Exercise 31. Evaluating Trigonometric Functions with a Calculator To use a calculator to evaluate trigonometric functions of angles measured in degrees, first set the calculator to degree mode and then proceed as demonstrated as follows. in Section 4.2. For instance, you can find values of cos and sec 28 28 Function cos 28 sec 28 a. b. Mode Degree Degree Calculator Keystrokes Display COS 28 ENTER 0.8829476 COS 28 x 1 ENTER 1.1325701 Throughout this text, angles are assumed to be measured in radians unless means noted otherwise. For example, sin 1 means the sine of 1 radian and the sine of 1 degree. sin 1 Example 6 Using a Calculator Use a calculator to evaluate sec5 40 12. Solution Begin by converting to decimal degree form. [Recall that 1 1 3600 1. 1 1 60 1 and 5 40 12 5 40 60 5.67 12 3600 sec 5.67. Then, use a calculator to evaluate Function sec5 40 12 sec 5.67 Calculator Keystrokes Display COS 5.67 x 1 ENTER 1.0049166 Now try Exercise 47. 333202_0403.qxd 12/7/05 11:03 |
AM Page 306 306 Chapter 4 Trigonometry Object Angle of elevation Horizontal Observer Observer Horizontal Angle of depression Object FIGURE 4.32 y Angle of elevation 78.3° x = 115 ft Not drawn to scale FIGURE 4.33 Applications Involving Right Triangles Many applications of trigonometry involve a process called solving right triangles. In this type of application, you are usually given one side of a right triangle and one of the acute angles and are asked to find one of the other sides, or you are given two sides and are asked to find one of the acute angles. In Example 7, the angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the horizontal, it is common to use the term angle of depression, as shown in Figure 4.32. Example 7 Using Trigonometry to Solve a Right Triangle A surveyor is standing 115 feet from the base of the Washington Monument, as shown in Figure 4.33. The surveyor measures the angle of elevation to the top of the monument as How tall is the Washington Monument? 78.3. Solution From Figure 4.33, you can see that tan 78.3 opp adj y x and x 115 where Washington Monument is y x tan 78.3 y 1154.82882 555 feet. is the height of the monument. So, the height of the Now try Exercise 63. Example 8 Using Trigonometry to Solve a Right Triangle An historic lighthouse is 200 yards from a bike path along the edge of a lake. A walkway to the lighthouse is 400 yards long. Find the acute angle between the bike path and the walkway, as illustrated in Figure 4.34. 200 yd θ 400 yd FIGURE 4.34 Solution From Figure 4.34, you can see that the sine of the angle 1 2 sin opp hyp 200 400. is Now you should recognize that 30. Now try Exercise 65. 333202_0403.qxd 12/7/05 11:03 AM Page 307 Section 4.3 Right Triangle Trigonometry 307 By now you are able to recognize that is the acute angle that Suppose, however, that you were given the 30 satisfies the equation equation sin 0.6 sin 1 2. and were asked to find the acute angle Because. and sin 30 1 2 0.5000 sin 45 1 2 0.7071 you might guess that In a later section, you will study a method by |
which a more precise value of can be determined. lies somewhere between and 45. 30 Example 9 Solving a Right Triangle Find the length of the skateboard ramp shown in Figure 4.35. c c 18.4° 4 ft FIGURE 4.35 Solution From Figure 4.35, you can see that sin 18.4 opp hyp 4 c. So, the length of the skateboard ramp is c 4 sin 18.4 4 0.3156 12.7 feet. Now try Exercise 67. 333202_0403.qxd 12/7/05 11:03 AM Page 308 308 Chapter 4 Trigonometry 4.3 Exercises VOCABULARY CHECK: 1. Match the trigonometric function with its right triangle definition. (a) Sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f) Cotangent (i) hypotenuse adjacent (ii) adjacent opposite (iii) hypotenuse opposite (iv) adjacent hypotenuse (v) opposite hypotenuse (vi) opposite adjacent In Exercises 2 and 3, fill in the blanks. 2. Relative to the angle, the three sides of a right triangle are the ________ side, the ________ side, and the ________. 3. An angle that measures from the horizontal upward to an object is called the angle of ________, whereas an angle that measures from the horizontal downward to an object is called the angle of ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, find the exact values of the six trigonometric functions of the angle shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.) In Exercises 9 –16, sketch a right triangle corresponding to the trigonometric function of the acute angle Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of.. 1. 6 θ 8 3. θ 41 9 2. 4. 13 θ 5 9. 11. 13. 15. sin 3 4 sec 2 tan 3 cot 3 2 10. 12. 14. 16. cos 5 7 cot 5 sec 6 csc 17 4 4 Function θ 4 In Exercises 5–8, find the exact values of the |
six trigonometric functions of the angle for each of the two triangles. Explain why the function values are the same. 5. 3 θ 1 2 6 θ 7. 1.25 1θ 5 4 θ 6. 8. θ 15 8 4 1 3 θ 7.5 θ 2 θ 6 17. sin 18. cos 19. tan 20. sec 21. cot 22. csc 23. cos 24. sin 25. cot 26. tan In Exercises 17–26, construct an appropriate triangle to complete the table. 0 ≤ ≤ 90, 0 ≤ ≤ /2 (deg) (rad) 30 45 Function Value 333202_0403.qxd 12/7/05 11:03 AM Page 309 In Exercises 27–32, use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions. 27. sin 60 3 2, cos 60 1 2 28. (a) tan 60 cos 30 (c) sin 30 1 2 csc 30 cos 30 (a) (c), tan 30 29. csc 13 2, sec (b) (d) sin 30 cot 60 3 3 (b) (d) 13 3 cot 60 cot 30 30. 31. 32. tan 26 (a) sin tan (c) sec 5, cos (a) cot90 (c) cos 1 3 sec (a) cot (c) tan 5 cot (a) tan90 (c) (b) (d) cos sec90 (b) (d) cot sin (b) (d) sin sin90 (b) (d) cos csc In Exercises 33–42, use trigonometric identities to transform the left side of the equation into the right side 0 < < /2. 33. tan cot 1 cos sec 1 tan cos sin cot sin cos 1 cos 1 cos sin2 1 sin 1 sin cos2 sec tan sec tan 1 sin2 cos2 2 sin2 1 sin cos sin cos tan cot tan csc sec csc2 34. 35. 36. 37. 38. 39. 40. 41. 42. In Exercises 43–52, use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) 43. (a) 44. (a) sin |
10 tan 23.5 (b) (b) cos 80 cot 66.5 Section 4.3 Right Triangle Trigonometry 309 45. (a) 46. (a) 47. (a) 48. (a) 49. (a) 50. (a) 51. (a) 52. (a) sin 16.35 cos 16 18 sec 42 12 cos 4 50 15 cot 11 15 sec 56 8 10 csc 32 40 3 sec9 5 20 32 (b) (b) (b) (b) (b) (b) (b) (b) csc 16.35 sin 73 56 csc 48 7 sec 4 50 15 tan 11 15 cos 56 8 10 tan 44 28 16 cot9 5 30 32 In Exercises 53–58, find the values of 0 < < 90 0 < < /2 and radians of a calculator. in degrees without the aid 53. (a) 54. (a) sin 1 2 2 2 cos 55. (a) sec 2 56. (a) tan 3 57. (a) 58. (a) csc 23 3 3 3 cot (b) csc 2 (b) tan 1 (b) (b) (b) cot 1 cos 1 2 2 2 sin (b) sec 2 In Exercises 59– 62, solve for x, y, or as indicated. r 59. Solve for x. 60. Solve for y. 30 30° x 18 y 60° 61. Solve for x. 62. Solve for r. 32 60° x r 20 45° 63. Empire State Building You are standing 45 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 86th floor (the observatory) is If the total height of the building is another 123 meters above the 86th floor, what is the approximate height of the building? One of your friends is on the 86th floor. What is the distance between you and your friend? 82. 333202_0403.qxd 12/7/05 11:03 AM Page 310 310 Chapter 4 Trigonometry 64. Height A six-foot person walks from the base of a broadcasting tower directly toward the tip of the shadow cast by the tower. When the person is 132 feet from the tower and 3 feet from the tip of the shadow, the person’s shadow starts to appear beyond the tower’s |
shadow. 68. Height of a Mountain In traveling across flat land, you notice a mountain directly in front of you. Its angle of 3.5. After you drive 13 miles elevation (to the peak) is 9. closer to the mountain, the angle of elevation is Approximate the height of the mountain. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the tower. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the tower? 65. Angle of Elevation You are skiing down a mountain with a vertical height of 1500 feet. The distance from the top of the mountain to the base is 3000 feet. What is the angle of elevation from the base to the top of the mountain? 66. Width of a River A biologist wants to know the width w of a river so in order to properly set instruments for the studying the pollutants in the water. From point C biologist walks downstream 100 feet and sights to point (see figure). From this sighting, it is determined that 54. How wide is the river? A, C w θ = 54° 100 ft A 67. Length A steel cable zip-line is being constructed for a competition on a reality television show. One end of the zip-line is attached to a platform on top of a 150-foot pole. The other end of the zip-line is attached to the top of a 23 5-foot stake. The angle of elevation to the platform is (see figure). θ = 23° 5 ft 150 ft (a) How long is the zip-line? (b) How far is the stake from the pole? (c) Contestants take an average of 6 seconds to reach the ground from the top of the zip-line. At what rate are contestants moving down the line? At what rate are they dropping vertically? 3.5° 13 mi 9° Not drawn to scale 69. Machine Shop Calculations A steel plate has the form of one-fourth of a circle with a radius of 60 centimeters. Two two-centimeter holes are to be drilled in the plate positioned as shown in the figure. Find the coordinates of the center of each hole. y 60 56 ( x y ), 2 2 30° 30° 30° ( x y ), 1 1 x 56 60 70. Machine Shop Calculations A tapered shaft has a |
diameter of 5 centimeters at the small end and is 15 centimeters long (see figure). The taper is Find the diameter of the large end of the shaft. 3. d 3° 5 cm d 15 cm 333202_0403.qxd 12/7/05 11:03 AM Page 311 Model It Synthesis Section 4.3 Right Triangle Trigonometry 311 71. Height A 20-meter line is used to tether a heliumfilled balloon. Because of a breeze, the line makes an angle of approximately with the ground. 85 (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the balloon? (d) The breeze becomes stronger and the angle the balloon makes with the ground decreases. How does this affect the triangle you drew in part (a)? (e) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures. Angle, 80 70 60 50 Height Angle, 40 30 20 10 Height (f) As the angle the balloon makes with the ground approaches how does this affect the height of the balloon? Draw a right triangle to explain your reasoning. 0, 20 72. Geometry Use a compass to sketch a quarter of a circle of radius 10 centimeters. Using a protractor, construct an angle of in standard position (see figure). Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle. By actual measurement, calculate the coordinates of the point of intersection and use these measurements to angle. approximate the six trigonometric functions of a x, y 20 y 10 (x, y) x 10 1 c m 0 20° True or False? statement is true or false. Justify your answer. In Exercises 73–78, determine whether the 73. 75. 77. sin 60 csc 60 1 sin 45 cos 45 1 sin 60 sin 30 sin 2 74. 76. sec 30 csc 60 cot2 10 csc2 10 1 78. tan52 tan25 79. Writing In right triangle trigonometry, explain why sin 30 1 2 regardless of the size of the triangle. 80. Think About It You are given only the value Is without finding the tan. it possible to find the value of measure of Explain.? sec 81. Exploration |
(a) Complete the table. 0.1 0.2 0.3 0.4 0.5 sin (b) Is or (c) As Explain. greater for sin approaches 0, how do in the interval and 0, 0.5? sin compare? 82. Exploration (a) Complete the table. 0 18 36 54 72 90 sin cos (b) Discuss the behavior of the sine function for in the range from 0 to 90. (c) Discuss the behavior of the cosine function for in the range from 0 to 90. (d) Use the definitions of the sine and cosine functions to explain the results of parts (b) and (c). Skills Review 83. In Exercises 83–86, perform the operations and simplify. x 2 12x 36 x 2 36 t 2 16 4t 2 12t 9 x 2 6x x 2 4x 12 2t 2 5t 12 9 4t 2 84. 85 4x 4 86. 1 4 1 3 x 12 x 333202_0404.qxd 12/7/05 11:05 AM Page 312 312 Chapter 4 Trigonometry 4.4 Trigonometric Functions of Any Angle What you should learn • Evaluate trigonometric functions of any angle. • Use reference angles to evaluate trigonometric functions. • Evaluate trigonometric functions of real numbers. Why you should learn it You can use trigonometric functions to model and solve real-life problems. For instance, in Exercise 87 on page 319, you can use trigonometric functions to model the monthly normal temperatures in New York City and Fairbanks, Alaska. Introduction In Section 4.3, the definitions of trigonometric functions were restricted to acute angles. In this section, the definitions are extended to cover any angle. If is an acute angle, these definitions coincide with those given in the preceding section. Definitions of Trigonometric Functions of Any Angle Let be an angle in standard position with of and r x2 y2 0. x, y a point on the terminal side sin y r tan y x sec r x, x 0, x 0 cos x r cot x y csc Because. functions are defined for any real value of However, if are undefined. For example, secant of y 0, Similarly, if x 0, 90 the cotangent and cosecant of are undefined. cannot be zero, it follows that the sine and cosine the tangent and is undefined. the |
tangent of Example 1 Evaluating Trigonometric Functions be a point on the terminal side of. Find the sine, cosine, and 3, 4 Let. tangent of Solution Referring to Figure 4.36, you can see that x 3, r x 2 y 2 32 42 25 5. y 4, and James Urbach/SuperStock So, you have the following. − ( 3, 4) r y 4 3 2 1 −3 −2 −1 FIGURE 4.36 θ 1 x sin y r cos x r tan y x 4 5 3 5 4 3 Now try Exercise 1. 333202_0404.qxd 12/7/05 11:05 AM Page 313 Section 4.4 Trigonometric Functions of Any Angle 313 π < <θ 3 2 0 < < < < 2θ π y y Quadrant II θ sin : + − θ cos : − θ tan : Quadrant I θ sin : + θ cos : + θ tan : + Quadrant III θ − sin : θ − cos : θ tan : + Quadrant IV θ − sin : θ cos : + θ − tan : x x FIGURE 4.37 y π 2 (0, 1) (−1, 0) π (1, 0) 0 x π 3 2 (0, −1) FIGURE 4.38 The signs of the trigonometric functions in the four quadrants can be determined easily from the definitions of the functions. For instance, because cos xr, which is in Quadrants I and IV. (Remember, is always positive.) In a similar manner, you can verify the results shown in Figure 4.37. is positive wherever it follows that cos r x > 0, Example 2 Evaluating Trigonometric Functions Given tan 5 4 and cos > 0, find sin and sec. Solution Note that tangent is negative and the cosine is positive. Moreover, using lies in Quadrant IV because that is the only quadrant in which the tan y x 5 4 and the fact that r 16 25 41 y and you have is negative in Quadrant IV, you can let y 5 and x 4. So, sin y r 5 41 0.7809 sec r x 41 4 1.6008. Now try Exercise 17. Example 3 Trigonometric Functions of Quadrant Angles Evaluate the cosine and tangent functions at the four quadrant |
angles 0, 3. 2 2,, and Solution To begin, choose a point on the terminal side of each angle, as shown in Figure 4.38. For each of the four points, and you have the following. cos 0 x r 1 1 1 0 cos x r 2 cos x r 0 1 1 1 1 r 1, tan 0 y x tan y x 2 tan y x cos 3 2 x r 0 1 0 tan 3 2 y x Now try Exercise 29. 0 1 1 0 0 x, y 1, 0 ⇒ undefined x, y 0, 1 0 0 1 1 0 x, y 1, 0 ⇒ undefined x, y 0, 1 333202_0404.qxd 12/7/05 11:05 AM Page 314 314 Chapter 4 Trigonometry Reference Angles (or less than ) can be determined from their values at corresponding acute angles called The values of the trigonometric functions of angles greater than 0 reference angles. 90 Definition of Reference Angle Let be an angle in standard position. Its reference angle is the acute angle formed by the terminal side of and the horizontal axis. Figure 4.39 shows the reference angles for in Quadrants II, III, and IV. Quadrant II θ Reference ′θ angle: θ Reference ′θ angle: θ Reference ′ θ angle: πθ θ ′ = − (radians) θ θ ′ = 180° − (degrees) FIGURE 4.39 Quadrant III ′ = − (radians) θ π θ ′ = − 180° (degrees) θ θ Quadrant IV θ θ ′ = 2 − (radians) θ θ ′ = 360° − (degrees) π Example 4 Finding Reference Angles Find the reference angle. 2.3 a. 300 b. c. 135 Solution a. Because 300 lies in Quadrant IV, the angle it makes with the -axis is x 360 300 60. Degrees Figure 4.40 shows the angle 300 2 1.5708 in Quadrant II and its reference angle is b. Because 2.3 lies between and its reference angle 3.1416, and 60. it follows that it is 2.3 0.8416. Radians c. First, determine that Figure 4.41 shows the angle 135 III. So, the reference angle is 225 180 2.3 and its reference angle |
225, is coterminal with which lies in Quadrant 2.3. 45. Degrees Figure 4.42 shows the angle 135 and its reference angle 45. y θ = 300° x ′ = 60° θ FIGURE 4.40 y ′ = − 2.3 π θ θ = 2.3 x FIGURE 4.41 y 225° and −135° are coterminal. 225° ′ = 45° θ x = −135° θ FIGURE 4.42 Now try Exercise 37. 333202_0404.qxd 12/7/05 11:05 AM Page 315 y (x, y) r = h y p opp θ ′ θ adj opp y, adj x FIGURE 4.43 Learning the table of values at the right is worth the effort because doing so will increase both your efficiency and your confidence. Here is a pattern for the sine function that may help you remember the values. 0 30 45 60 90 sin Reverse the order to get cosine values of the same angles. Section 4.4 Trigonometric Functions of Any Angle 315 Trigonometric Functions of Real Numbers To see how a reference angle is used to evaluate a trigonometric function, consider the point as shown in Figure 4.43. By definition, you know that on the terminal side of x, y, sin y r and tan y x. For the right triangle with acute angle have x sin opp hyp y r and sides of lengths x and y, you and tan opp adj y x. sin tan and tan are equal, except possibly in sign. The same is So, it follows that and for the other four trigonometric functions. In all true for and cases, the sign of the function value can be determined by the quadrant in which sin lies. Evaluating Trigonometric Functions of Any Angle To find the value of a trigonometric function of any angle : 1. Determine the function value for the associated reference angle. 2. Depending on the quadrant in which lies, affix the appropriate sign to the function value. By using reference angles and the special angles discussed in the preceding section, you can greatly extend the scope of exact trigonometric values. For means that you know the function instance, knowing the function values of values of all angles for which is a reference angle. For convenience, the table below shows the exact values of the trigonometric functions of special angles and quadrant angles |
. 30 30 Trigonometric Values of Common Angles (degrees) 0 30 45 60 (radians) sin cos tan 90 2 1 0 180 0 270 3 2 1 1 0 1 3 Undef. 0 Undef. 333202_0404.qxd 12/7/05 11:05 AM Page 316 316 Chapter 4 Trigonometry Example 5 Using Reference Angles Evaluate each trigonometric function. a. cos 4 3 b. tan210 c. csc 11 4 Solution a. Because 43 43 3, negative in Quadrant III, so lies in Quadrant III, as shown in Figure 4.44. Moreover, the reference angle is the cosine is cos 4 3 cos 3 1 2. b. Because 210 360 150, 150. is coterminal with the as shown in Figure 4.45. Finally, because the tangent is 210 reference angle it follows that the So, is second-quadrant angle 180 150 30, negative in Quadrant II, you have tan210 tan 30 3 3 114 2 34,. c. Because with the second-quadrant angle 34 4, positive in Quadrant II, you have it follows that is coterminal the reference angle is as shown in Figure 4.46. Because the cosecant is 34. So, 114 csc 11 4 csc 4 1 sin4 2. y y y ′ = 30° θ θ = π 4 3 x ′θ = π 4 x θ = π 11 210° FIGURE 4.44 FIGURE 4.45 FIGURE 4.46 Now try Exercise 51. 333202_0404.qxd 12/7/05 11:05 AM Page 317 Section 4.4 Trigonometric Functions of Any Angle 317 Example 6 Using Trigonometric Identities Let be an angle in Quadrant II such that by using trigonometric identities. sin 1 3. Find (a) cos and (b) tan Solution a. Using the Pythagorean identity sin2 cos2 1, you obtain 2 1 3 cos2 1 Substitute 1 3 for sin. cos 2 1 1 9 8 9. Because cos < 0 in Quadrant II, you can use the negative root to obtain cos 8 9 22 3. b. Using the trigonometric identity tan sin cos, you obtain Substitute for sin and cos. tan 13 223 1 22 2 4. Now try Exercise 59. You can use a calculator to evaluate trigon |
ometric functions, as shown in the next example. Example 7 Using a Calculator Use a calculator to evaluate each trigonometric function. a. cot 410 b. sin7 c. sec 9 Solution Function Mode Calculator Keystrokes Display a. b. c. cot 410 sin7 sec 9 Degree Radian TAN SIN 410 7 x 1 ENTER ENTER 0.8390996 0.6569866 Radian COS 9 x 1 ENTER 1.0641778 Now try Exercise 69. 333202_0404.qxd 12/7/05 11:05 AM Page 318 318 Chapter 4 Trigonometry 4.4 Exercises VOCABULARY CHECK: In Exercises 1– 6, let be an angle in standard position, with x, y a point on the terminal side of and rx2 y2 0. 1. sin ________ 3. 5. tan ________ x r ________ 2. 4. 6. ________ r y sec ________ x y ________ 7. The acute positive angle that is formed by the terminal side of the angle and the horizontal axis is called the ________ angle of and is denoted by. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, determine the exact values of the six trigonometric functions of the angle. 9. 3.5, 6.8 10. 31 2, 73 4 1. (a) 2. (a) (4, 3) θ y y θ (−12, −5) 3. (a) y θ − 3, −1 ) ( 4. (a) y (3, 1b) θ (8, 15)− (b) − ( 1, 1) (b) − ( 4, 1) (b) θ θ θ x x x x (4, 4)− In Exercises 5–10, the point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. 5. 7. 7, 24 4, 10 6. 8. 8, 15 5, 2 In Exercises 11–14, state the quadrant in which lies. 11. 12. 13. 14. sin < 0 sin > 0 sin > 0 sec > 0 and and and and cos < 0 cos |
> 0 tan < 0 cot < 0 In Exercises 15–24, find the values of the six trigonometric functions of with the given constraint. Function Value Constraint 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. sin 3 5 cos 4 5 tan 15 8 cos 8 17 cot 3 csc 4 sec 2 sin 0 cot tan is undefined. is undefined. lies in Quadrant II. lies in Quadrant III. sin < 0 tan < 0 cos > 0 cot < 0 sin > 0 sec 1 2 ≤ ≤ 32 ≤ ≤ 2 lies on the given In Exercises 25–28, the terminal side of line in the specified quadrant. Find the values of the six trigonometric functions of by finding a point on the line. Line y x y 1 3x 2x y 0 4x 3y 0 25. 26. 27. 28. Quadrant II III III IV 333202_0404.qxd 12/7/05 11:05 AM Page 319 In Exercises 29–36, evaluate the trigonometric function of the quadrant angle. Section 4.4 Trigonometric Functions of Any Angle 319 In Exercises 65–80, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) 29. sin 31. sec 33. sin 3 2 2 35. csc 30. csc 3 2 32. sec 34. cot 36. cot 2 In Exercises 37–44, find the reference angle in standard position. and 65. 67. 69. 71. 73. 75. 77. 79., and sketch sin 10 cos110 tan 304 sec 72 tan 4.5 tan 9 sin0.65 cot11 8 66. 68. 70. 72. 74. 76. 78. 80. sec 225 csc330 cot 178 tan188 cot 1.35 tan 9 sec 0.29 csc15 14 37. 39. 41. 203 245 2 3 43. 3.5 38. 40. 42. 44. 309 145 7 4 11 3 In Exercises 45–58, evaluate the sine, cosine, and tangent of the angle without using a calculator. 46. 48. 50. 52. 54. 56. 300 405 840 4 2 10 3 45. 47. 49. 51. 53. 55. 57. |
58. 225 750 150 4 3 6 11 4 3 2 25 4 In Exercises 59–64, find the indicated trigonometric value in the specified quadrant. Function sin 3 5 cot 3 tan 3 2 csc 2 cos 5 8 sec 9 4 59. 60. 61. 62. 63. 64. Quadrant IV II III IV I III Trigonometric Value cos sin sec cot sec tan In Exercises 81–86, find two solutions of the equation. Give and in radians your answers in degrees 0 ≤ < 2. Do not use a calculator. 0 ≤ < 360 81. (a) 82. (a) 83. (a) sin 1 2 2 cos 2 csc 23 3 84. (a) 85. (a) sec 2 tan 1 86. (a) sin 3 2 (b) (b) sin 1 2 2 2 cos (b) cot 1 (b) (b) sec 2 cot 3 3 2 (b) sin Model It 87. Data Analysis: Meteorology The table shows the monthly normal temperatures (in degrees Fahrenheit) and for selected months for New York City F. Fairbanks, Alaska (Source: National Climatic Data Center) N Month January April July October December New York City, N Fairbanks, F 33 52 77 58 38 10 32 62 24 6 (a) Use the regression feature of a graphing utility to find a model of the form y a sinbt c d for each city. Let represent the month, with corresponding to January. t t 1 333202_0404.qxd 12/7/05 11:05 AM Page 320 320 Chapter 4 Trigonometry Model It (co n t i n u e d ) (b) Use the models from part (a) to find the monthly normal temperatures for the two cities in February, March, May, June, August, September, and November. (c) Compare the models for the two cities. 88. Sales A company that produces snowboards, which are seasonal products, forecasts monthly sales over the next 2 years to be FIGURE FOR 92 Synthesis d θ 6 mi Not drawn to scale S 23.1 0.442t 4.3 cos t 6 S is measured in thousands of units and where the time in months, with Predict sales for each of the following months. is representing January 2006. t 1 t (a) February 2006 (b) February 2007 (c) June 2006 (d) June 2007 89 |
. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring is given by yt 2 cos 6t True or False? the statement is true or false. Justify your answer. In Exercises 93 and 94, determine whether 93. In each of the four quadrants, the signs of the secant function and sine function will be the same. 94. To find the reference angle for an angle in degrees), find 0 ≤ 360n ≤ 360. reference angle. integer the The difference n such 360n (given that is the 95. Writing Consider an angle in standard position with centimeters, as shown in the figure. Write a short x, y, 0 r 12 paragraph describing the changes in the values of sin, 90. to increases continuously from cos, tan and as y is the displacement (in centimeters) and t where time (in seconds). Find the displacement when (a) (b) and (c) t 1 2. t 1 4, is the t 0, y 90. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by y t 2et cos 6t y is the displacement (in centimeters) and t where time (in seconds). Find the displacement when (a) (b) and (c) t 1 2. t 1 4, is the t 0, 12 cm θ (x, y) x 96. Writing Explain how reference angles are used to find 91. Electric Circuits The current I (in amperes) when the trigonometric functions of obtuse angles. 100 volts is applied to a circuit is given by I 5e2t sin t Skills Review t where is the time (in seconds) after the voltage is applied. second after the voltage Approximate the current at is applied. t 0.7 92. Distance An airplane, flying at an altitude of 6 miles, is on a flight path that passes directly over an observer (see is the angle of elevation from the observer to figure). If d the plane, find the distance from the observer to the plane 30, 90, 120. when (a) and (c) (b) In Exercises 97–106, graph the function. Identify the domain and any intercepts and asymptotes of the function. 97. 99. 101. y x2 3x 4 f x x3 8 f x x 7 x2 4x 4 103 |
. 105. y 2x1 y ln x4 98. 100. 102. y 2x2 5x gx x4 2x2 3 hx x2 1 x 5 y 3 x1 2 104. 106. y log10 x 2 333202_0405.qxd 12/7/05 11:06 AM Page 321 Section 4.5 Graphs of Sine and Cosine Functions 321 4.5 Graphs of Sine and Cosine Functions What you should learn • Sketch the graphs of basic sine and cosine functions. • Use amplitude and period to help sketch the graphs of sine and cosine functions. • Sketch translations of the graphs of sine and cosine functions. • Use sine and cosine functions to model real-life data. Why you should learn it Sine and cosine functions are often used in scientific calculations. For instance, in Exercise 73 on page 330, you can use a trigonometric function to model the airflow of your respiratory cycle. Basic Sine and Cosine Curves In this section, you will study techniques for sketching the graphs of the sine and cosine functions. The graph of the sine function is a sine curve. In Figure 4.47, the black portion of the graph represents one period of the function and is called one cycle of the sine curve. The gray portion of the graph indicates that the basic sine curve repeats indefinitely in the positive and negative directions. The graph of the cosine function is shown in Figure 4.48. Recall from Section 4.2 that the domain of the sine and cosine functions is the set of all real numbers. Moreover, the range of each function is the interval 1, 1, Do you see how this information is consistent with the basic graphs shown in Figures 4.47 and 4.48? and each function has a period of 2. Range: − ≤ ≤y 1 1 FIGURE 4.47 y = sin 1 y 1 −1 Period: 2π y = cos x π π 2 π 3 2 2π π 5 2 Period: 2π Note in Figures 4.47 and 4.48 that the sine curve is symmetric with respect to the origin, whereas the cosine curve is symmetric with respect to the -axis. These properties of symmetry follow from the fact that the sine function is odd and the cosine function is even. y © Karl Weatherly/Corbis Range: − ≤ ≤y |
1 1 − π 3 2 −π FIGURE 4.48 333202_0405.qxd 12/7/05 11:06 AM Page 322 322 Chapter 4 Trigonometry To sketch the graphs of the basic sine and cosine functions by hand, it helps to note five key points in one period of each graph: the intercepts, maximum points, and minimum points (see Figure 4.49). y y Intercept Maximum Intercept ), 1 ( y π 2 Minimum Intercept Intercept Minimum Intercept = sin x (0, 1) Maximum y = cos x π (, 0) ( π 3 2 ), 1− (0, 0) Quarter period Half period π Period: 2 Three-quarter period x π(2, 0) Full period Quarter period π Period)− π Half period Three-quarter period Maximum π (2, 1) x Full period FIGURE 4.49 Example 1 Using Key Points to Sketch a Sine Curve Sketch the graph of y 2 sin x on the interval, 4. Solution Note that y 2 sin x 2sin x indicates that the -values for the key points will have twice the magnitude of those on the graph of into four equal parts to get Divide the period the key points for y sin x. y 2 sin x. 2 y Intercept Maximum Intercept, 2, 2, 0, 0, 0, Minimum, 2, 3 2 Intercept and 2, 0 By connecting these key points with a smooth curve and extending the curve in you obtain the graph shown in Figure both directions over the interval 4.50., 4, Te c h n o l o g y When using a graphing utility to graph trigonometric functions, pay special attention to the viewing window you use. For instance, y [sin10x]/10 try graphing the standard viewing window in radian mode. What do you observe? Use the zoom feature to find a viewing window that displays a good view of the graph. in y 3 2 1 − π 2 −2 FIGURE 4.50 y = 2 sin x y = sin x 3π 2 5π 2 7π 2 x Now try Exercise 35. 333202_0405.qxd 12/7/05 11:06 AM Page 323 Section 4.5 Graphs of Sine and Cosine Functions 323 Amplitude and Period In the remainder of this section you will study the graphic effect of each of the constants in equations of the forms d a, and c, |
b, y d a sinbx c and y d a cosbx c. A quick review of the transformations you studied in Section 1.7 should help in this investigation. y a sin x in The constant factor a stretch or vertical shrink of the basic sine curve. If is stretched, and if y a sin x graph of a absolute value of function acts as a scaling factor—a vertical a > 1, the basic sine curve the basic sine curve is shrunk. The result is that the and 1. The The range of the a < 1, ranges between instead of between y a sin x. is the amplitude of the function a ≤ y ≤ a. y a sin x a and a > 0 1 a for is Definition of Amplitude of Sine and Cosine Curves The amplitude of and between the maximum and minimum values of the function and is given by represents half the distance y a cos x y a sin x Amplitude a. y = 3 cos x Example 2 Scaling: Vertical Shrinking and Stretching y = cos x On the same coordinate axes, sketch the graph of each function. a. y 1 2 cos x b. y 3 cos x x 2π y 1 = cos 2 x Solution a. Because the amplitude of minimum value is get the key points 1 2. y 1 is Divide one cycle, 2 cos x 1 2, 0 ≤ x ≤ 2, the maximum value is and the into four equal parts to 1 2 Maximum Intercept 0,, 0,, 2 1 2 Minimum,, 1 2 Intercept, 0, 3 2 Maximum. 2, 1 2 and y 3 −1 −2 −3 FIGURE 4.51 Exploration y cos bx Sketch the graph of b 1 2, and 3. How does for 2, b the value of affect the graph? How many complete cycles occur between 0 and for each value of b? 2 b. A similar analysis shows that the amplitude of y 3 cos x is 3, and the key points are Maximum Intercept Minimum, 0,, 3, 0, 3, 2 Intercept, 0, 3 2 Maximum and 2, 3. The graphs of these two functions are shown in Figure 4.51. Notice that the graph y 1 of and the graph of y 3 cos x is a vertical shrink of the graph of is a vertical stretch of the graph of y cos x. y cos x 2 cos x Now try Exercise 37. 333202_0405.qxd 12/7/05 11: |
06 AM Page 324 324 Chapter 4 Trigonometry y y = 3 cos x y = −3 cos x 3 1 You know from Section 1.7 that the graph of y f x. y f x For instance, the graph of is a reflection in the y 3 cos x is a x -axis of the graph of reflection of the graph of y a sin x Because y a sin bx that y 3 cos x, completes one cycle from to x 0 completes one cycle from as shown in Figure 4.52. x 0 to x 2b. x 2, it follows π− π 2π x Period of Sine and Cosine Functions Let be a positive real number. The period of b is given by y a sin bx and y a cos bx −3 FIGURE 4.52 Period 2. b Note that if 0 < b < 1, Exploration Sketch the graph of y sinx c c 4, 0, 4. where How does the value of affect the graph? and c In general, to divide a period-interval into four equal parts, successively add “period/4,” starting with the left endpoint of the interval. For instance, for the period-interval 6, 2 you would successively add of length 23, 6 23 4 6, 0, 6, 3, and as the -values for the key to get 2 points on the graph. x the period of represents a horizontal stretching of the graph of the period of of the graph of cosx cos x y a sin bx y a sin x. are used to rewrite the function. is less than If 2 b is negative, the identities y a sin x. y a sin bx is greater than Similarly, if 2 and b > 1, and represents a horizontal shrinking and sinx sin x Example 3 Scaling: Horizontal Stretching Sketch the graph of y sin x 2. Solution The amplitude is 1. Moreover, because b 1 2, the period is 2 b 2 1 2 4. Substitute for b. Now, divide the period-interval 2, and 3 to obtain the key points on the graph. 0, 4 into four equal parts with the values, Intercept Maximum Intercept 0, 0,, 1, 2, 0, Minimum 3, 1, Intercept 4, 0 and The graph is shown in Figure 4.53. y = sin x y 1 y = sin x 2 − π π −1 Period: 4 π FIGURE |
4.53 Now try Exercise 39. x 333202_0405.qxd 12/7/05 11:06 AM Page 325 Section 4.5 Graphs of Sine and Cosine Functions 325 Translations of Sine and Cosine Curves The constant c y a sinbx c in the general equations and y a cosbx c creates a horizontal translation (shift) of the basic sine and cosine curves. y a sinbx c, Comparing you find that the graph of y a sinbx c By solving for bx c 0 you can find the interval for one cycle to be completes one cycle from bx c 2. y a sin bx with to x, Left endpoint Right endpoint Period This implies that the period of y a sin bx is shifted by an amount y a sinbx c is The number cb. 2b, cb and the graph of is the phase shift. Graphs of Sine and Cosine Functions y a sinbx c The graphs of b > 0. ) characteristics. (Assume and y a cosbx c have the following Amplitude a Period 2 b The left and right endpoints of a one-cycle interval can be determined by solving the equations and bx c 2. bx c 0 Example 4 Horizontal Translation Sketch the graph of y 1 2 sinx. 3 Solution 1 The amplitude is and the period is 2 2. and By solving the equations 3, 73 you see that the interval Dividing this interval into four equal parts produces the key points corresponds to one cycle of the graph. Intercept Maximum, 3 5, 6, 0, 1 2 Intercept, 0, 4 3 Minimum, 11, 1 2 6 Intercept, 0. 7 3 and y 1 2 y = sin 2π π 5 3 π 8 3 x Period: 2 π The graph is shown in Figure 4.54. FIGURE 4.54 Now try Exercise 45. 333202_0405.qxd 12/7/05 11:06 AM Page 326 326 Chapter 4 Trigonometry y = −3 cos(2 x + 4 ) π π Example 5 Horizontal Translation y 3 2 −3 −2 Period 1 FIGURE 4.55 Sketch the graph of y 3 cos2x 4. Solution The amplitude is 3 and the period is x 1 2x 4 0 22 1. By solving the equations 2x 4 x 2 and 2x 4 2 2x 2 x 1 |
2, 1 you see that the interval Dividing this interval into four equal parts produces the key points corresponds to one cycle of the graph. Minimum 2, 3, Intercept Maximum Intercept 7 3, 0, 4 2 5 4, 0,, 3, Minimum and 1, 3. The graph is shown in Figure 4.55. Now try Exercise 47. The final type of transformation is the vertical translation caused by the constant d in the equations y d a sinbx c and y d a cosbx c. d The shift is units upward for words, the graph oscillates about the horizontal line x -axis. d > 0 d and units downward for d < 0. In other instead of about the y d y = 2 + 3 cos 2x Example 6 Vertical Translation y 5 Sketch the graph of y 2 3 cos 2x. − π 1 −1 π x Period π FIGURE 4.56 Solution. The amplitude is 3 and the period is The key points over the interval, 1,, 2,, 2, 0, 5,, 5. and 4 2 3 4 0, are The graph is shown in Figure 4.56. Compared with the graph of the graph of is shifted upward two units. y 2 3 cos 2x f x 3 cos 2x, Now try Exercise 53. 333202_0405.qxd 12/7/05 11:06 AM Page 327 Section 4.5 Graphs of Sine and Cosine Functions 327 Mathematical Modeling Sine and cosine functions can be used to model many real-life situations, including electric currents, musical tones, radio waves, tides, and weather patterns. Time, t Depth, y Example 7 Finding a Trigonometric Model Midnight 2 A.M. 4 A.M. 6 A.M. 8 A.M. 10 A.M. Noon 3.4 8.7 11.3 9.1 3.8 0.1 1.2 Changing Tides y 12 10 FIGURE 4.57 4 A.M. 8 A.M. Time Noon t Throughout the day, the depth of water at the end of a dock in Bar Harbor, Maine varies with the tides. The table shows the depths (in feet) at various times during the morning. (Source: Nautical Software, Inc.) a. Use a trigonometric function to model the data. b. Find the depths at 9 A.M. and 3 P.M. c. A boat needs |
at least 10 feet of water to moor at the dock. During what times in the afternoon can it safely dock? Solution a. Begin by graphing the data, as shown in Figure 4.57. You can use either a sine or cosine model. Suppose you use a cosine model of the form y a cosbt c d. The difference between the maximum height and the minimum height of the graph is twice the amplitude of the function. So, the amplitude is a 1 2 maximum depth minimum depth 1 2 11.3 0.1 5.6. The cosine function completes one half of a cycle between the times at which the maximum and minimum depths occur. So, the period is p 2time of min. depth time of max. depth b 2p 0.524. 210 4 which implies that midnight, consider the left endpoint to be 1 because the average depth is 2 you can model the depth with the function given by 11.3 0.1 5.7, Because high tide occurs 4 hours after Moreover, So, c 2.094. it follows that cb 4, d 5.7. 12 so 12 (14.7, 10) (17.3, 10) b. The depths at 9 A.M. and 3 P.M. are as follows. y 5.6 cos0.524t 2.094 5.7. y = 10 0 24 0 y = 5.6 cos(0.524t − 2.094) + 5.7 FIGURE 4.58 y 5.6 cos0.524 9 2.094 5.7 0.84 foot y 5.6 cos0.524 15 2.094 5.7 10.57 feet 9 A.M. 3 P.M. c. To find out when the depth y 10 is at least 10 feet, you can graph the model with the line using a graphing utility, as shown in Figure 4.58. Using the intersect feature, you can determine that the depth is at least 10 feet between t 17.3. 2:42 P.M. and 5:18 P.M. t 14.7 y Now try Exercise 77. 333202_0405.qxd 12/7/05 11:06 AM Page 328 328 Chapter 4 Trigonometry 4.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. One period of a sine or cosine function function is |
called one ________ of the sine curve or cosine curve. 2. The ________ of a sine or cosine curve represents half the distance between the maximum and minimum values of the function. 3. The period of a sine or cosine function is given by ________. 4. For the function given by y a sinbx c, c b represents the ________ ________ of the graph of the function. 5. For the function given by y d a cosbx c, d represents a ________ ________ of the graph of the function. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–14, find the period and amplitude. 1. y 3 sin 2x 2. y 2 cos 3x 3 π x 3. y 5 2 cos x 2 4. y 3 sin x 3 y 3 −2 −3 x π2 y 4 π π− −2 −4 x 5. y 1 2 sin x 3 6. y 3 2 cos x 2 y 1 π 2 −1 x y 2 − π −2 x π 7. y 2 sin x 9. 11. y 3 sin 10x 2x y 1 2 3 cos 8. 10. 12. y cos 2x 3 3 sin 8x x 4 cos y 1 y 5 2 In Exercises 15–22, describe the relationship between the graphs of and Consider amplitude, period, and shifts. g. 13. 14. y 1 4 y 2 3 sin 2x cos x 10 15. 17. 19. 21. f f x sin x gx sinx f x cos 2x gx cos 2x f x cos x gx cos 2x f x sin 2x gx 3 sin 2x 16. 18. 20. 22. f x cos x gx cosx f x sin 3x gx sin3x f x sin x gx sin 3x f x cos 4x gx 2 cos 4x In Exercises 23–26, describe the relationship between the graphs of and Consider amplitude, period, and shifts. g. f 23. 252 −3 3 2 1 −2 −3 −2 π 24. y x x 26. 3 2 g −2 −2 π π 2 −2 x x 333202_0405.qxd 12/7/05 11:06 |
AM Page 329 In Exercises 27–34, graph and on the same set of coordinate axes. (Include two full periods.) g f 27. f x 2 sin x gx 4 sin x 29. 31. 33. sin f x cos x gx 1 cos x x f x 1 2 2 gx 3 1 2 f x 2 cos x gx 2 cosx sin x 2 28. f x sin x x 3 gx sin 30. f x 2 cos 2x gx cos 4x 32. f x 4 sin x gx 4 sin x 3 34. f x cos x gx cosx In Exercises 35–56, sketch the graph of the function. (Include two full periods.) 65. 35. 37. 39. y 3 sin x y 1 3 cos x x 2 y cos 41. y cos 2x 43. 45. y sin 2x 3 y sinx 4 47. y 3 cosx 49. 51. 53. 54. 55. y 2 sin 2x 3 y 2 1 10 cos 60x y 3 cosx 3 4 y 4 cosx 4 cosx 2 y 2 3 4 36. 38. y 1 4 sin x y 4 cos x 40. y sin 4x 42. y sin x 4 44. y 10 cos x 6 46. y sinx y 4 cosx 48. 4 t 12 50. y 3 5 cos 52. y 2 cos x 3 56. y 3 cos6x 57. In Exercises 57– 62, use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. y 2 sin4x y cos2x y 3 cosx 2 y 4 sin2 3 2 2 1 x 60. 58. 59. 3 2 Section 4.5 Graphs of Sine and Cosine Functions 329 y 0.1 sinx 10 y 1 100 sin 120t 61. 62. Graphical Reasoning for the function matches the figure. f x a cos x d In Exercises 63– 66, find d and f such that the graph of a 63. y y 4 1 −1 −2 10 8 6 4 − π −2 64 66. − π f π f y −3 −4 1 −1 −2 −5 x x Graphical Reasoning for the function f matches the figure. f x a sinbx c In Exercises 67–70, find c such that the graph |
of and a, b, 67. 69. y y f 1 −3 3 2 1 −2 −3 f 68. x π − π 70. x π y 3 2 1 −3 y 3 2 −2 −3 f π f 2 4 x x In Exercises 71 and 72, use a graphing utility to graph [2, 2]. and Use the graphs to find y2. y1 real numbers such that in the interval x y2 y1 71. y1 y2 sin x 1 2 72. cos x 1 y1 y2 333202_0405.qxd 12/7/05 11:06 AM Page 330 330 Chapter 4 Trigonometry 73. Respiratory Cycle For a person at rest, the velocity (in liters per second) of air flow during a respiratory cycle (the time from the beginning of one breath to the beginning of v the next) is given by t 3 seconds). (Inhalation occurs when occurs when v 0.85 sin v < 0. ), where is the time (in t v > 0, and exhalation (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Sketch the graph of the velocity function. 74. Respiratory Cycle After exercising for a few minutes, a person has a respiratory cycle for which the velocity of air flow is approximated by v 1.75 sin t 2, where t is the time (in seconds). (Inhalation occurs when exhalation occurs when v < 0. ) v > 0, and (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Sketch the graph of the velocity function. 75. Data Analysis: Meteorology The table shows the maxiand t 1 (Source: National Climatic mum daily high temperatures for Tallahassee Chicago corresponding to January. Data Center) T (in degrees Fahrenheit) for month with C t, (c) Use a graphing utility to graph the data points and the model for the temperatures in Chicago. How well does the model fit the data? (d) Use the models to estimate the average maximum temperature in each city. Which term of the models did you use? Explain. (e) What is the period of each model? Are the periods what you expected? Explain. (f) Which city has the greater variability in temperature throughout the year? |
Which factor of the models determines this variability? Explain. 76. Health The function given by approximates the blood pressure t mercury at time (in seconds) for a person at rest. P 100 20 cos P (in millimeters) of 5t 3 (a) Find the period of the function. (b) Find the number of heartbeats per minute. 77. Piano Tuning When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up a wave y 0.001 sin 880t, motion that can be approximated by where is the time (in seconds). t (a) What is the period of the function? (b) The frequency f is given by f 1p. What is the frequency of the note? Month, t Tallahassee, T Chicago, C Model It 1 2 3 4 5 6 7 8 9 10 11 12 63.8 67.4 74.0 80.0 86.5 90.9 92.0 91.5 88.5 81.2 72.9 65.8 29.6 34.7 46.1 58.0 69.9 79.2 83.5 81.2 73.9 62.1 47.1 34.4 78. Data Analysis: Astronomy The percent moon’s face that is illuminated on day 2007, where the table. of the of the year represents January 1, is shown in (Source: U.S. Naval Observatory) x 1 y x x 3 11 19 26 32 40 y 1.0 0.5 0.0 0.5 1.0 0.5 (a) A model for the temperature in Tallahassee is given by Tt 77.90 14.10 cost 6 3.67. Find a trigonometric model for Chicago. (b) Use a graphing utility to graph the data points and the model for the temperatures in Tallahassee. How well does the model fit the data? (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. (c) Add the graph of your model in part (b) to the scatter plot. How well does the model fit the data? (d) What is the period of the model? (e) Estimate the moon’s percent illumination for March 12, 2007. 333202_0405.qxd 12/7/05 11:06 AM Page 331 79. Fuel |
Consumption The daily consumption lons) of diesel fuel on a farm is modeled by C 30.3 21.6 sin2t 365 10.9 C (in gal- is the time (in days), with t where January 1. t 1 corresponding to (a) What is the period of the model? Is it what you expected? Explain. (b) What is the average daily fuel consumption? Which term of the model did you use? Explain. (c) Use a graphing utility to graph the model. Use the graph to approximate the time of the year when consumption exceeds 40 gallons per day. 80. Ferris Wheel A Ferris wheel is built such that the height (in feet) above ground of a seat on the wheel at time (in t h seconds) can be modeled by ht 53 50 sin 10 t. 2 (a) Find the period of the model. What does the period tell you about the ride? (b) Find the amplitude of the model. What does the ampli- tude tell you about the ride? (c) Use a graphing utility to graph one cycle of the model. Section 4.5 Graphs of Sine and Cosine Functions 331 87. Exploration Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials sin x x x3 3! cos x 1 x 2 2! x5 5! x4 4! and where x is in radians. (a) Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Use a graphing utility to graph the cosine function and its polynomial approximation in the same viewing window. How do the graphs compare? (c) Study the patterns in the polynomial approximations of the sine and cosine functions and predict the next term in each. Then repeat parts (a) and (b). How did the accuracy of the approximations change when an additional term was added? 88. Exploration Use the polynomial approximations for the sine and cosine functions in Exercise 87 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain. (a) sin 1 2 (b) sin 1 (c) sin 6 4 cos Synthesis (d) cos0.5 (e) cos 1 ( |
f) True or False? statement is true or false. Justify your answer. In Exercises 81– 83, determine whether the Skills Review 81. The graph of the function given by translates the graph of the right so that the two graphs look identical. f x sin x f x sinx 2 exactly one period to 82. The function given by y 1 is twice that of the function given by 83. The graph of y sinx 2 y cos x x in the -axis. 2 cos 2x has an amplitude that y cos x. is a reflection of the graph of y d a sinbx c, 84. Writing Use a graphing utility to graph the function given a, and Write a paragraph describing the changes in the by c,b, graph corresponding to changes in each constant. for several different values of d. Conjecture In Exercises 85 and 86, graph and on the same set of coordinate axes. Include two full periods. Make a conjecture about the functions. f x sin x, gx cosx 85. g f 2 f x sin x, gx cosx 86. 2 In Exercises 89–92, use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of a logarithm. log10 x 2 t 3 t 1 log2 ln z x2x 3 z2 1 92. 89. 91. 90. ln In Exercises 93–96, write the expression as the logarithm of a single quantity. 93. 95. 96. log10 x log10 y 1 2 ln 3x 4 ln y ln 2x 2 ln x 3 ln x 1 2 94. 2 log2 x log2 xy 97. Make a Decision To work an extended application analyzing the normal daily maximum temperature and normal precipitation in Honolulu, Hawaii, visit this text’s website at college.hmco.com. (Data Source: NOAA) 333202_0406.qxd 12/8/05 8:43 AM Page 332 332 Chapter 4 Trigonometry 4.6 Graphs of Other Trigonometric Functions What you should learn • Sketch the graphs of tangent functions. • Sketch the graphs of cotangent functions. • Sketch the graphs of secant and cosecant functions. • Sketch the graphs of damped trigonometric functions. Graph of the Tangent Function Recall that the tangent function is odd |
. That is, the graph of from the identity cos x 0. which Consequently, is symmetric with respect to the origin. You also know that the tangent is undefined for values at x ± 2 ±1.5708. tan x sin xcos x Two such values are y tan x tanx tan x. Why you should learn it tan x Undef. 1255.8 14.1 x 2 1.57 1.5 4 1 0 0 4 1 1.5 1.57 2 14.1 1255.8 Undef. Trigonometric functions can be used to model real-life situations such as the distance from a television camera to a unit in a parade as in Exercise 76 on page 341. y tan x increases without bound as approaches x As indicated in the table, tan from from the right. So, the left, and decreases without bound as approaches x 2, as the graph of and, shown in Figure 4.59. Moreover, because the period of the tangent function is is an integer. The vertical asymptotes also occur when domain of the tangent function is the set of all real numbers other than x 2 n, and the range is the set of all real numbers. x 2 x 2 has vertical asymptotes at x 2 n, where n x 2 PERIOD: x DOMAIN: ALL 2, RANGE: VERTICAL ASYMPTOTES: x 2 n n y y = tan x 3 2 1 Photodisc/Getty Images π 33 FIGURE 4.59 y a tanbx c Sketching the graph of is similar to sketching the graph of y a sinbx c in that you locate key points that identify the intercepts and asymptotes. Two consecutive vertical asymptotes can be found by solving the equations bx c 2 and bx c. 2 The midpoint between two consecutive vertical asymptotes is an -intercept of the graph. The period of the function is the distance between two consecutive vertical asymptotes. The amplitude of a tangent function is not defined. After plotting the asymptotes and the -intercept, plot a few additional points between the two asymptotes and sketch one cycle. Finally, sketch one or two additional cycles to the left and right. y a tanbx c x x 333202_0406.qxd 12/8/05 8:43 AM Page 333 Section 4.6 Graphs of |
Other Trigonometric Functions 333 Example 1 Sketching the Graph of a Tangent Function Sketch the graph of y tan x 2. y y = tan x 2 3 2 1 −π π x π3 −3 FIGURE 4.60 y y = −3 tan 2x x π 4 π 2 π3 4 6 −2 −4 −6 − π3 4 − π 2 − π 4 FIGURE 4.61 Solution By solving the equations and. and you can see that two consecutive vertical asymptotes occur at x Between these two asymptotes, plot a few points, including the -intercept, as shown in the table. Three cycles of the graph are shown in Figure 4.60. x 2 tan x 2 Undef. 1 2 1 Undef. 0 0 Now try Exercise 7. Example 2 Sketching the Graph of a Tangent Function Sketch the graph of y 3 tan 2x. Solution By solving the equations 2x x 2 4 and 2x x 2 4 you can see that two consecutive vertical asymptotes occur at and x 4. Between these two asymptotes, plot a few points, including the -intercept, as shown in the table. Three cycles of the graph are shown in Figure 4.61. x x 4 x 4 8 3 tan 2x Undef. 3 8 4 3 Undef. 0 0 Now try Exercise 9. By comparing the graphs in Examples 1 and 2, you can see that the graph of increases between consecutive vertical asymptotes when In is a reflection in the -axis of the graph for a > 0. y a tanbx c a > 0, other words, the graph for and decreases between consecutive vertical asymptotes when a < 0. a < 0 x 333202_0406.qxd 12/8/05 8:43 AM Page 334 334 Chapter 4 Trigonometry Graph of the Cotangent Function The graph of the cotangent function is similar to the graph of the tangent function. It also has a period of However, from the identity. y cot x cos x sin x Te c h n o l o g y Some graphing utilities have difficulty graphing trigonometric functions that have vertical asymptotes. Your graphing utility may connect parts of the graphs of tangent, cotangent, secant, and cosecant functions that are not supposed to be connected. To eliminate this problem |
, change the mode of the graphing utility to dot mode. y y = 2 cot x 3 3 2 1 −2 π π 3π π 4 π 6 x FIGURE 4.63 x n, is zero, you can see that the cotangent function has vertical asymptotes when is an integer. The graph of the cotangent funcwhich occurs at tion is shown in Figure 4.62. Note that two consecutive vertical asymptotes of the bx c 0 graph of and can be found by solving the equations y a cotbx c bx c. where sin x n y y = cot x 3 2 1 −π − π 2 π 2 π π3 2 π2 x PERIOD: DOMAIN: ALL RANGE: VERTICAL ASYMPTOTES: x n, x n FIGURE 4.62 Example 3 Sketching the Graph of a Cotangent Function Sketch the graph of y 2 cot x 3. Solution By solving the equations and. you can see that two consecutive vertical asymptotes occur at Between these two asymptotes, plot a few points, including the -intercept, as shown in the table. Three cycles of the graph are shown in Figure 4.63. Note that the distance between consecutive asymptotes. the period is x 0 x 3, and cot x 3 Undef. 2 0 2 Undef. Now try Exercise 19. 333202_0406.qxd 12/8/05 8:43 AM Page 335 Section 4.6 Graphs of Other Trigonometric Functions 335 Graphs of the Reciprocal Functions The graphs of the two remaining trigonometric functions can be obtained from the graphs of the sine and cosine functions using the reciprocal identities sec x 1 cos x csc x 1 sin x and. y For instance, at a given value of x. the -coordinate of cos Of course, when x, exist. Near such values of of the tangent function. In other words, the graphs of is the reciprocal of the reciprocal does not the behavior of the secant function is similar to that the -coordinate of sec cos x 0, x, x y tan x sin x cos x and sec x 1 cos x x 2 n, have vertical asymptotes at is zero at these -values. Similarly, x cot x cos x sin x and csc x 1 |
sin x where n is an integer, and the cosine have vertical asymptotes where sin x 0 —that is, at x n. To sketch the graph of a secant or cosecant function, you should first make a y csc x, Then take reciprocals of the -coordinates to This procedure is used to obtain the sketch of its reciprocal function. For instance, to sketch the graph of first sketch the graph of obtain points on the graph of graphs shown in Figure 4.64. y csc x. y sin x. y y 3 y = sin x y = csc x −π −1 −2 −3 y = sec x π π 2 x π 2 y = cos x x n 2 PERIOD: DOMAIN: ALL, 1 RANGE: VERTICAL ASYMPTOTES: SYMMETRY: ORIGIN FIGURE 4.64 1, x n 2 PERIOD: x n DOMAIN: ALL 2, 1 RANGE: VERTICAL ASYMPTOTES: SYMMETRY: 1, x 2 -AXIS y n In comparing the graphs of the cosecant and secant functions with those of the sine and cosine functions, note that the “hills” and “valleys” are interchanged. For example, a hill (or maximum point) on the sine curve corresponds to a valley (a relative minimum) on the cosecant curve, and a valley (or minimum point) on the sine curve corresponds to a hill (a relative maximum) on the cosecant curve, as shown in Figure 4.65. Additionally, -intercepts of the sine and cosine functions become vertical asymptotes of the cosecant and secant functions, respectively (see Figure 4.65). x Cosecant: relative minimum Sine: minimum x π2 y 4 3 2 1 −1 −2 −3 −4 π Sine: maximum Cosecant: relative maximum FIGURE 4.65 333202_0406.qxd 12/8/05 8:43 AM Page 336 336 Chapter 4 Trigonometry y = 2 csc sin x + ( π 4 ) Example 4 Sketching the Graph of a Cosecant Function 4 3 1 Sketch the graph of y 2 cscx. 4 Solution Begin by sketching the graph of π � |
� 2 x y 2 sinx. 4 For this function, the amplitude is 2 and the period is 2. By solving the equations FIGURE 4.66 and you can see that one cycle of the sine function corresponds to the interval from x 4 The graph of this sine function is represented by the gray curve in Figure 4.66. Because the sine function is zero at the midpoint and endpoints of this interval, the corresponding cosecant function x 74. to y 2 cscx 4 sinx 4 1 2 has vertical asymptotes at the cosecant function is represented by the black curve in Figure 4.66. x 4, x 34, x 74, etc. The graph of Now try Exercise 25. Example 5 Sketching the Graph of a Secant Function Sketch the graph of y sec 2x. y = sec 2x y y = cos 2x 3 −1 −2 −3 −π − π 2 x π π 2 Solution Begin by sketching the graph of Figure 4.67. Then, form the graph of x Note that the -intercepts of, 0,, 0, y cos 2x 3 4 4 4, 0,... y cos 2x, y sec 2x as indicated by the gray curve in as the black curve in the figure. correspond to the vertical asymptotes x 3 4 x x,, 4 4,... of the graph of y sec 2x. is y sec 2x. Moreover, notice that the period of y cos 2x and FIGURE 4.67 Now try Exercise 27. 333202_0406.qxd 12/8/05 8:43 AM Page 337 Section 4.6 Graphs of Other Trigonometric Functions 337 Damped Trigonometric Graphs A product of two functions can be graphed using properties of the individual functions. For instance, consider the function f x x sin x as the product of the functions value and the fact that sin x ≤ 1, y x y sin x. and you have 0 ≤ xsin x ≤ x. Using properties of absolute Consequently, y = x x ≤ x sin x ≤ x which means that the graph of y x. Furthermore, because f x x sin x lies between the lines y x and π x and f x x sin x ±x at x 2 n f x x sin x 0 at x n y = −x y π3 π 2 π − |
π π−2 π−3 f(x) = x sin x FIGURE 4.68 at x 2 n x Do you see why the graph of f x x sin x touches the lines y ±x and why the graph has -intercepts at Recall that the sine function is equal to 1 at 32, 52,... of 2, 2, odd multiples and is equal to 0 at multiples of. 2 3,... x n?, f(x) = e−x sin 3x y 6 4 −4 −6 y = e−e−x and x f the graph of x has -intercepts at f x x sin x, tion touches the line x n. the factor x y x A sketch of or the line f and is shown in Figure 4.68. In the func- x 2 n y x at is called the damping factor. Example 6 Damped Sine Wave Sketch the graph of f x ex sin 3x. Solution Consider f x y ex as the product of the two functions y sin 3x and each of which has the set of real numbers as its domain. For any real number you know that that ex sin 3x ≤ ex, x, which means sin 3x ≤ 1. ex ≥ 0 and So, ex ≤ ex sin 3x ≤ ex. Furthermore, because f x ex sin 3x ±ex at x 6 n 3 f x ex sin 3x 0 at x n 3 y ex the graph of and has intercepts at touches the curves x n3. and A sketch is shown in Figure 4.69. at f y ex x 6 n3 FIGURE 4.69 Now try Exercise 65. 333202_0406.qxd 12/8/05 8:43 AM Page 338 338 Chapter 4 Trigonometry Figure 4.70 summarizes the characteristics of the six basic trigonometric functions. y 2 1 y = sin x −π π− 2 π 2 π π 3 2 −2 DOMAIN: ALL REALS RANGE: PERIOD: 2 1, 1 y y = csc x = 1 sin x 3 2 1 y 2 y = cos x x π− π x π 2 −1 −2 DOMAIN: ALL REALS RANGE: PERIOD: 2 1, 1 y y = sec x = 1 cos x 3 y y = tan π5 2 x 2, DOMAIN: ALL RANGE |
: PERIOD: n y y = cot x = 1 tan x 3 2 1 −π π π 2 x π2 − π2 x −2 −3 x n DOMAIN: ALL, 1 RANGE: 2 PERIOD: FIGURE 4.70 1, x n 2, 1 1, DOMAIN: ALL RANGE: PERIOD: 2 x n DOMAIN: ALL RANGE: PERIOD:, W RITING ABOUT MATHEMATICS Combining Trigonometric Functions Recall from Section 1.8 that functions can be combined arithmetically. This also applies to trigonometric functions. For each of the functions hx x sin x and hx cos x sin 3x (a) identify two simpler functions and table to show how to obtain the numerical values of of and (c) use graphs of and gx, f x and g g f f that comprise the combination, (b) use a hx from the numerical values to show how may be formed. h Can you find functions f x d a sinbx c and gx d a cosbx c such that f x gx 0 for all x? 333202_0406.qxd 12/8/05 8:43 AM Page 339 Section 4.6 Graphs of Other Trigonometric Functions 339 4.6 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The graphs of the tangent, cotangent, secant, and cosecant functions all have ________ asymptotes. 2. To sketch the graph of a secant or cosecant function, first make a sketch of its corresponding ________ function. 3. For the functions given by f x gx sin x, gx is called the ________ factor of the function f x. 4. The period of is ________. 5. The domain of is all real numbers such that ________. y tan x y cot x y sec x y csc x 6. The range of 7. The period of is ________. is ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, match the function with its graph. State the period of the function. [The graphs are labeled (a), (b), (c), (d), ( |
e), and (f).] (b) (d) (f) (a) (c) (e) − 3π 3 −3 y 3 x 2 x π 3 2 In Exercises 7–30, sketch the graph of the function. Include two full periods. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 2 sec x y 1 3 tan x y tan 3x y 1 y csc x y sec x 1 x 2 y csc y cot y 1 x 2 2 sec 2x x 4 y tan y csc x y 2 secx cscx y 1 4 4 8. 10. 12. 14. 16. 18. 20. 22. y 1 4 tan x y 3 tan x y 1 4 sec x y 3 csc 4x y 2 sec 4x 2 x 3 y csc y 3 cot y 1 x 2 2 tan x 24. y tanx 26. 28. 30. y csc2x y sec x 1 y 2 cotx 2 In Exercises 31– 40, use a graphing utility to graph the function. Include two full periods. x 1 31. y tan x 3 32. y tan 2x 1. y sec 2x 3. 5. y 1 2 y 1 2 cot x sec x 2 2. y tan x 2 4. y csc x 6. y 2 sec x 2 33. 35. 37. 39. y 2 sec 4x y tanx y csc4x y 0.1 tanx 4 4 4 34. 36. y sec x y 1 4 cotx y 2 sec2x secx 2 38. 40. y 1 3 2 2 333202_0406.qxd 12/8/05 8:43 AM Page 340 340 Chapter 4 Trigonometry In Exercises 41– 48, use a graph to solve the equation on the interval [2, 2]. 41. 42. tan x 1 tan x 3 43. cot x 3 3 44. 45. 46. 47. 48. cot x 1 sec x 2 sec x 2 csc x 2 csc x 23 3 In Exercises 49 and 50, use the graph of the function to determine whether the function is even, odd, or neither. 49. f x sec x 50. f x tan x − π 51. Graphical Reasoning Consider the functions given by f |
x 2 sin x and gx 1 2 csc x on the interval 0,. g (a) Graph and f in the same coordinate plane. (b) Approximate the interval in which f > g. (c) Describe the behavior of each of the functions as approaches How is the behavior of behavior of as approaches? x. f g x related to the 52. Graphical Reasoning Consider the functions given by f x tan x 2 and gx 1 2 sec x 2 on the interval 1, 1. (a) Use a graphing utility to graph viewing window. f and g in the same (b) Approximate the interval in which f < g. (c) Approximate the interval in which 2f < 2g. How does the result compare with that of part (b)? Explain. In Exercises 53–56, use a graphing utility to graph the two equations in the same viewing window. Use the graphs to determine whether the expressions are equivalent. Verify the results algebraically. 53. 54. 55. 56. y1 y1 y1 y1 sin x csc x, sin x sec x, cos x sin x y2, 1 tan x y2 y2 cot x sec2 x 1, tan2 x y2 In Exercises 57– 60, match the function with its graph. Describe the behavior of the function as approaches zero. [The graphs are labeled (a), (b), (c), and (d).] x (a) (c) x π 2 (b) (d) x π − π y y 2 −1 −2 −3 −4 −5 −6 4 2 −2 −4 y 4 2 y −4 4 3 2 1 −1 −2 x π 2 π 3 2 x π 57. 58. 59. 60. f x x cos x f x x sin x gx x sin x gx x cos x In Exercises 61–64, graph the functions and Use the graphs to make a conjecture about the f Conjecture g. relationship between the functions. f x sin x cosx 61. f x sin x cosx f x sin2 x, gx 1 2 62. 63. gx 0, 2, 2 1 cos 2x gx 2 sin x 64. f x cos2 x 2, gx 1 2 1 cos x In Exercises 65–68, use a grap |
hing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as increases without bound. x gx ex 22 sin x f x 2x4 cos x 65. 67. 66. 68. f x ex cos x hx 2x24 sin x Exploration In Exercises 69–74, use a graphing utility to graph the function. Describe the behavior of the function as approaches zero. x 69. y 6 x cos x, x > 0 70. y 4 x sin 2x, x > 0 333202_0406.qxd 12/8/05 8:43 AM Page 341 71. gx sin x x 73. f x sin 1 x 72. f x 1 cos x x 74. hx x sin 1 x d 75. Distance A plane flying at an altitude of 7 miles above a radar antenna will pass directly over the radar antenna (see figure). Let be the ground distance from the antenna to x the point directly under the plane and let be the angle of is positive as the elevation to the plane from the antenna. ( d x plane approaches the antenna.) Write and graph the function over the interval d as a function of 0 < x <. x d 7 mi Not drawn to scale 76. Television Coverage A television camera is on a reviewing platform 27 meters from the street on which a parade will be passing from left to right (see figure). Write from the camera to a particular unit in the the distance and graph the function parade as a function of the angle 2 < x < 2. over the interval as (Consider negative when a unit in the parade approaches from the left.) x, d x Not drawn to scale 27 m d x Camera Model It 77. Predator-Prey Model The population of coyotes (a predator) at time (in months) in a region is estimated to be C t C 5000 2000 sin t 12 and the population of rabbits (its prey) is estimated to be R Section 4.6 Graphs of Other Trigonometric Functions 341 Model It (co n t i n u e d ) R 25,000 15,000 cos t 12. (a) Use a graphing utility to graph both models in the same viewing window. Use the window setting 0 ≤ t ≤ 100. (b) Use the graphs of the models in part (a) to explain the oscillations in the size of each population. (c) The cycles of each population follow a |
periodic pattern. Find the period of each model and describe several factors that could be contributing to the cyclical patterns. 78. Sales The projected monthly sales (in thousands of units) of lawn mowers (a seasonal product) are modeled by S 74 3t 40 cost6, t is the time (in months), with corresponding to January. Graph the sales function over 1 year. where t 1 S 79. Meterology The normal monthly high temperatures H (in degrees Fahrenheit) for Erie, Pennsylvania are approximated by Ht 54.33 20.38 cos t 6 15.69 sin t 6 and the normal monthly low temperatures mated by L are approxi- Lt 39.36 15.70 cos t 6 14.16 sin t 6 is the time (in months), with t where to January (see figure). Atmospheric Administration) corresponding (Source: National Oceanic and ( 80 60 40 20 H(t) L(t) 1 2 3 4 5 7 8 6 Month of year 9 10 11 12 t (a) What is the period of each function? (b) During what part of the year is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is northernmost in the sky around June 21, but the graph shows the warmest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun. 333202_0406.qxd 12/8/05 8:43 AM Page 342 342 Chapter 4 Trigonometry 80. Harmonic Motion An object weighing pounds is suspended from the ceiling by a steel spring (see figure). The weight is pulled downward (positive direction) from its equilibrium position and released. The resulting motion of the weight is described by the function W y 1 2 et4 cos 4t, t > 0 y where seconds). is the distance (in feet) and t is the time (in Equilibrium 86. Approximation Using calculus, it can be shown that the tangent function can be approximated by the polynomial tan x x 2x3 3! 16x 5 5! x is in radians. Use a graphing utility to graph the where tangent function and its polynomial approximation in the same viewing window. How do the graphs compare? 87. Approximation Using calculus, it can be shown that the secant function can be approximated by the polynomial sec x 1 x 2 2! 5x4 4! x |
where is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare? y 88. Pattern Recognition (a) Use a graphing utility to graph each function. (a) Use a graphing utility to graph the function. (b) Describe the behavior of the displacement function for t. increasing values of time Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 81 and 82, determine whether 81. The graph of y csc x can be obtained on a calculator by graphing the reciprocal of y sin x. 82. The graph of y sec x can be obtained on a calculator by graphing a translation of the reciprocal of y sin x. 83. Writing Describe the behavior of f x tan x as x approaches 2 from the left and from the right. 84. Writing Describe the behavior of f x csc x as x approaches from the left and from the right. 85. Exploration Consider the function given by f x x cos x. (a) Use a graphing utility to graph the function and verify that there exists a zero between 0 and 1. Use the graph to approximate the zero. 1, cosxn1 generate a sequence For example, x1, x2, where x0 xn. (b) Starting with x3,..., 1 x0 cosx0 x1 cosx1 x2 cosx2 x3 What value does the sequence approach? y1 y2 sin x 1 4 sin x 1 4 3 3 sin 3x sin 3x 1 5 sin 5x (b) Identify the pattern started in part (a) and find a that continues the pattern one more term. y3. function Use a graphing utility to graph y3 (c) The graphs in parts (a) and (b) approximate the periodic that is a better function in the figure. Find a function approximation. y4 y 1 x 3 Skills Review In Exercises 89–92, solve the exponential equation. Round your answer to three decimal places. 89. 91. e2x 54 300 1 ex 100 90. 92. 83x 98 1 0.15 365 365t 5 In Exercises 93–98, solve the logarithmic equation. Round your answer to three decimal places. 93. 95. ln3x 2 73 lnx2 1 3.2 |
log8 x log8 97. 98. log6 x log6 x 1 1 3 x2 1 log6 64x 94. 96. ln14 2x 68 ln x 4 5 333202_0407.qxd 12/7/05 11:10 AM Page 343 4.7 Inverse Trigonometric Functions Section 4.7 Inverse Trigonometric Functions 343 What you should learn • Evaluate and graph the inverse sine function. • Evaluate and graph the other inverse trigonometric functions. • Evaluate and graph the compositions of trigonometric functions. Why you should learn it You can use inverse trigonometric functions to model and solve real-life problems. For instance, in Exercise 92 on page 351, an inverse trigonometric function can be used to model the angle of elevation from a television camera to a space shuttle launch. NASA When evaluating the inverse sine function, it helps to remember the phrase “the arcsine of is the angle (or number) whose sine is x.” x Inverse Sine Function Recall from Section 1.9 that, for a function to have an inverse function, it must be one-to-one—that is, it must pass the Horizontal Line Test. From Figure 4.71, x you can see that y yield the same -value. does not pass the test because different values of y sin x y 1 −1 π− y = sin x π x sin x has an inverse function on this interval. FIGURE 4.71 However, if you restrict the domain to the interval 2 ≤ x ≤ 2 (corresponding to the black portion of the graph in Figure 4.71), the following properties hold. 2, 2, 2, 2, the function y sin x y sin x takes on its full range of values, is increasing. 1. On the interval 2. On the interval 1 ≤ sin x ≤ 1. 3. On the interval 2, 2, y sin x 2 ≤ x ≤ 2, is one-to-one. y sin x So, on the restricted domain function called the inverse sine function. It is denoted by has a unique inverse y arcsin x or y sin1 x. sin1 x is consistent with the inverse function notation x The notation The x arcsin notation (read as “the arcsine of ”) comes from the association of a central angle with its intercepted arc length on a unit circle. So, arcsin means |
x are the angle (or arc) whose sine is denotes the inverse commonly used in mathematics, so remember that sine function rather than lie in the interval 2 ≤ arcsin x ≤ 2. sin1 x x The values of arcsin y arcsin x 1sin x. The graph of is shown in Example 2. Both notations, arcsin x sin1 x, and x. f 1x. Definition of Inverse Sine Function The inverse sine function is defined by y arcsin x if and only if sin y x 1 ≤ x ≤ 1 and 2 ≤ y ≤ 2. The domain of y arcsin x is where 1, 1, and the range is 2, 2. 333202_0407.qxd 12/7/05 11:10 AM Page 344 344 Chapter 4 Trigonometry Example 1 Evaluating the Inverse Sine Function As with the trigonometric functions, much of the work with the inverse trigonometric functions can be done by exact calculations rather than by calculator approximations. Exact calculations help to increase your understanding of the inverse functions by relating them to the right triangle definitions of the trigonometric functions. y π 2 π ( ) 1, 2 2( 2, ) π 4 (0, 0) −( 1, − 2 FIGURE 4.72 ( 1 2, π 6 ) 1 x y = arcsin If possible, find the exact value. arcsin 1 2 sin1 3 2 b. a. c. sin1 2 Solution a. Because sin arcsin 1 2 1 2 6 for 2 ≤ y ≤, 2 it follows that. 6 Angle whose sine is 1 2 b. Because sin 3 3 2 for 2 ≤ y ≤, 2 it follows that sin1 3 2. 3 Angle whose sine is 32 c. It is not possible to evaluate because there is no angle whose sine is 2. Remember that the domain of the inverse sine function is 1, 1. when y sin1 x x 2 Now try Exercise 1. Example 2 Graphing the Arcsine Function Sketch a graph of y arcsin x. Solution By definition, 2 ≤ y ≤ 2. 2, 2, of values. Then plot the points and draw a smooth curve through the points. are equivalent for their graphs are the same. From the interval in the second equation to make a table the equations So, you can assign values to y arcsin x sin y x and y y 2 4 6 x sin |
arcsin x is shown in Figure 4.72. Note that it is the The resulting graph for reflection (in the line ) of the black portion of the graph in Figure 4.71. Be sure you see that Figure 4.72 shows the entire graph of the inverse sine function. and the Remember that the domain of range is the closed interval y arcsin x 2, 2. is the closed interval 1, 1 y x Now try Exercise 17. 333202_0407.qxd 12/7/05 11:10 AM Page 345 Section 4.7 Inverse Trigonometric Functions 345 Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval shown in Figure 4.73. 0 ≤ x ≤, as y y = cos x π− −1 π π 2 x π 2 cos x has an inverse function on this interval. FIGURE 4.73 Consequently, on this interval the cosine function has an inverse function—the inverse cosine function—denoted by y arccos x or y cos1 x. y tan x Similarly, you can define an inverse tangent function by restricting the domain of The following list summarizes the definitions of the three most common inverse trigonometric functions. The remaining three are defined in Exercises 101–103. 2, 2. to the interval Definitions of the Inverse Trigonometric Functions Function Domain y arcsin x if and only if sin y x 1 ≤ x ≤ 1 Range 2 ≤ y ≤ y arccos x if and only if cos arctan x if and only if tan The graphs of these three inverse trigonometric functions are shown in Figure 4.74 = arcsin x 1 −1 y = arccos x π 2 y = arctan x −2 −1 1 2 x − π 2 1, 1 DOMAIN: RANGE: 2, 2 FIGURE 4.74 −1 x 1 − π 2 1, 1 DOMAIN: RANGE: 0,, DOMAIN: RANGE: 2, 2 333202_0407.qxd 12/8/05 8:25 AM Page 346 346 Chapter 4 Trigonometry Example 3 Evaluating Inverse Trigonometric Functions Find the exact value. a. arccos 2 2 c. arctan 0 Solution a. Because b. cos11 d. tan11 cos4 22,. and 4 lies in 0,, it follows that Angle whose |
cosine is 22 arccos 2 2 4 b. Because and lies in 0,, it follows that cos 1, cos11. tan 0 0, arctan 0 0. c. Because and 0 lies in Angle whose cosine is 2, 2, 1 it follows that Angle whose tangent is 0 d. Because tan4 1, and 4 lies in 2, 2, it follows that tan11. 4 Angle whose tangent is 1 Now try Exercise 11. Example 4 Calculators and Inverse Trigonometric Functions Use a calculator to approximate the value (if possible). a. b. c. arctan8.45 sin1 0.2447 arccos 2 Solution It is important to remember that the domain of the inverse sine function and the inverse cosine 1, 1, function is as indicated in Example 4(c). a. b. c. Mode Function arctan8.45 From the display, it follows that Radian Calculator Keystrokes TAN1 8.45 ENTER arctan8.45 1.453001. sin1 0.2447 From the display, it follows that Radian SIN1 0.2447 ENTER sin1 0.2447 0.2472103. arccos 2 In real number mode, the calculator should display an error message because the domain of the inverse cosine function is 1, 1. Radian ENTER 2 COS1 Now try Exercise 25. In Example 4, if you had set the calculator to degree mode, the displays would have been in degrees rather than radians. This convention is peculiar to calculators. By definition, the values of inverse trigonometric functions are always in radians. 333202_0407.qxd 12/7/05 11:10 AM Page 347 Section 4.7 Inverse Trigonometric Functions 347 Compositions of Functions Recall from Section 1.9 that for all tions have the properties f f 1x x and f 1 f x x. x in the domains of and f f 1, inverse func- Inverse Properties of Trigonometric Functions 2 ≤ y ≤ 2, If 1 ≤ x ≤ 1 and sinarcsin x x then arcsinsin y y. and If 1 ≤ x ≤ 1 and 0 ≤ y ≤, then cosarccos x x and arccoscos y y. If x is a real number and tanarctan x x 2 < y < 2, then and arctantan y y. |
Keep in mind that these inverse properties do not apply for arbitrary values of and For instance, x y. arcsinsin arcsin1 3 2 2 3. 2 In other words, the property arcsinsin y y is not valid for values of outside the interval y 2, 2. Example 5 Using Inverse Properties If possible, find the exact value. a. tanarctan5 arcsinsin b. 5 3 c. coscos1 Solution a. Because 5 applies, and you have lies in the domain of the arctan function, the inverse property tanarctan5 5. 53 2 ≤ y ≤ 2. b. In this case, 5 3 2 3 does not lie within the range of the arcsine function, However, is coterminal with 53 which does lie in the range of the arcsine function, and you have arcsinsin arcsinsin 5 3 coscos1 3. 3 c. The expression is not defined because cos1 is not defined. Remember that the domain of the inverse cosine function is Now try Exercise 43. 1, 1. 333202_0407.qxd 12/7/05 11:10 AM Page 348 348 Chapter 4 Trigonometry y 3 2 2 3 − 2 5= u = arccos 2 3 2 Angle whose cosine is FIGURE 4.75 2 3 y 52 − −32 = 4 ( ) ( ( u = arcsin − 3 5 −3 5 x x Angle whose sine is FIGURE 4.76 3 5 Example 6 shows how to use right triangles to find exact values of compositions of inverse functions. Then, Example 7 shows how to use right triangles to convert a trigonometric expression into an algebraic expression. This conversion technique is used frequently in calculus. Example 6 Evaluating Compositions of Functions Find the exact value. tanarccos a. 2 3 b. cosarcsin 3 5 Solution a. If you let then cos u 2 3. u arccos 2 3, Because quadrant angle. You can sketch and label angle Consequently, tanarccos 2 3 u arcsin3 tan u opp adj, sin u 3 5. 5 2 then. 5 b. If you let fourth-quadrant angle. You can sketch and label angle 4.76. Consequently, cosarcsin 3 5 cos u adj hyp 4 5. cos u u is positive, is a firstas shown in Figure 4.75. u Because sin u u is negative, is a as shown |
in Figure u Now try Exercise 51. Example 7 Some Problems from Calculus Write each of the following as an algebraic expression in x. a. sinarccos 3x, 0 ≤ x ≤ 1 3 b. cotarccos 3x3x)2 Solution If you let u arccos 3x, then cos u 3x, where 1 ≤ 3x ≤ 1. Because u = arccos 3x 3x Angle whose cosine is FIGURE 4.77 3x cos u adj hyp 3x 1 you can sketch a right triangle with acute angle this triangle, you can easily convert each expression to algebraic form. as shown in Figure 4.77. From u, a. b. sinarccos 3x sin u opp hyp cotarccos 3x cot u adj opp 1 9x 2, 0 ≤ x ≤ 3x 1 9x Now try Exercise 59. In Example 7, similar arguments can be made for values lying in the x- interval 1 3, 0. 333202_0407.qxd 12/7/05 11:10 AM Page 349 Section 4.7 Inverse Trigonometric Functions 349 4.7 Exercises VOCABULARY CHECK: Fill in the blanks. Function Alternative Notation Domain 1. y arcsin x 2. __________ y arctan x 3. __________ y cos1 x __________ __________ 1 ≤ x ≤ 1 __________ Range 2 ≤ y ≤ 2 __________ __________ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–16, evaluate the expression without using a calculator. In Exercises 35 and 36, determine the missing coordinates of the points on the graph of the function. 1. 3. 5. 7. 9. 11. 13. 15. arcsin 1 2 arccos 1 2 3 3 arctan cos1 3 2 arctan3 arccos 1 2 sin1 3 2 tan1 0 2. 4. 6. 8. 10. 12. 14. 16. arcsin 0 arccos 0 arctan1 sin1 2 2 arctan 3 2 2 arcsin 3 3 tan1 cos1 1 35. y π 2 π 4 −3 −2 − 3, ( ) y = arctan x ( π ) 4 36. |
y π π 4 (−1, 1 ( − 2, ) ) −2 −1 y = arccos x ( 1 π ) 6, x 2 In Exercises 37–42, use an inverse trigonometric function to write as a function of 37. 38. x. In Exercises 17 and 18, use a graphing utility to graph y x and cally that the domain of properly.) f, g, in the same viewing window to verify geometrig (Be sure to restrict is the inverse function of f. f f x sin x, f x tan x, 17. 18. gx arcsin x gx arctan x 39. In Exercises 19–34, use a calculator to evaluate the expression. Round your result to two decimal places. 41. 19. 21. 23. 25. 27. 29. 31. 33. arccos 0.28 arcsin0.75 arctan3 sin1 0.31 arccos0.41 arctan 0.92 arcsin 3 4 tan1 7 2 20. 22. 24. 26. 28. 30. 32. 34. arcsin 0.45 arccos0.7 arctan 15 cos1 0.26 arcsin0.125 arctan 2.8 arccos1 tan195 3 7 x x + 2 40. 42. x θ 4 θ 10 2x θ x + 3 In Exercises 43–48, use the properties of inverse trigonometric functions to evaluate the expression. 43. 45. sinarcsin 0.3 cosarccos0.1 47. arcsinsin 3 44. tanarctan 25 sinarcsin0.2 7 2 46. 48. arccoscos 333202_0407.qxd 12/7/05 11:10 AM Page 350 350 Chapter 4 Trigonometry In Exercises 49–58, find the exact value of the expression. (Hint: Sketch a right triangle.) sinarctan 3 49. 50. 4 51. 53. 55. 57. costan1 2 cosarcsin 5 secarctan3 sinarccos2 13 5 3 52. 54. 56. 58. secarcsin 4 5 sincos1 5 5 cscarctan 5 tanarcsin3 cotarctan 5 12 4 8 In Exercises 59–68, write an algebraic expression that is equivalent |
to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.) In Exercises 77–82, sketch a graph of the function. 77. 78. 79. 80. 81. y 2 arccos x gt arccost 2 f x) arctan 2x f x arctan x 2 hv tanarccos v 82. f x arccos x 4 In Exercises 83– 88, use a graphing utility to graph the function. 59. 61. 63. 65. 66. 67. cotarctan x cosarcsin 2x sinarccos x tanarccos x 3 cotarctan 1 x cscarctan cosarcsin 68. x 2 x h r 60. 62. 64. sinarctan x secarctan 3x secarcsinx 1 83. 84. 85. 86. 87. 88. f x 2 arccos2x f x arcsin4x f x arctan2x 3 f x 3 arctanx f x sin12 3 cos1 1 f x 2 In Exercises 69 and 70, use a graphing utility to graph and g in the same viewing window to verify that the two functions are equal. Explain why they are equal. Identify any asymptotes of the graphs. f 69. f x sinarctan 2x, gx f x tanarccos 70., x 2 gx 2x 1 4x2 4 x 2 x In Exercises 71–74, fill in the blank. 71. arctan arcsin, x 0 9 x 36 x 2 6 72. arcsin arccos, 0 ≤ x ≤ 6 73. arccos 74. arccos 3 x 2 2x 10 x 2 2 arcsin arctan, x 2 ≤ 2 In Exercises 75 and 76, sketch a graph of the function and f x arcsin x. compare the graph of with the graph of g 75. gx arcsinx 1 76. gx arcsin x 2 In Exercises 89 and 90, write the function in terms of the sine function by using the identity A cos t B sin t A2 B2 sint arctan. A B Use a graphing utility to graph both forms of the function. What does the graph imply? f t 3 cos 2t 3 sin 2t f t 4 cos t 3 sin |
t 90. 89. 91. Docking a Boat A boat is pulled in by means of a winch located on a dock 5 feet above the deck of the boat (see figure). Let be the angle of elevation from the boat to the winch and let be the length of the rope from the winch to the boat. s 5 ft s θ (a) Write as a function of s. (b) Find when s 40 feet and s 20 feet. 333202_0407.qxd 12/7/05 11:10 AM Page 351 92. Photography A television camera at ground level is filming the lift-off of a space shuttle at a point 750 meters from the launch pad (see figure). Let be the angle of elevation to the shuttle and let be the height of the shuttle. s Section 4.7 Inverse Trigonometric Functions 351 94. Granular Angle of Repose Different types of granular substances naturally settle at different angles when stored is called the angle of in cone-shaped piles. This angle repose (see figure). When rock salt is stored in a coneshaped pile 11 feet high, the diameter of the pile’s base is about 34 feet. (Source: Bulk-Store Structures, Inc.) 11 ft θ 17 ft s θ 750 m Not drawn to scale (a) Write as a function of s. (a) Find the angle of repose for rock salt. (b) How tall is a pile of rock salt that has a base diameter of 40 feet? 95. Granular Angle of Repose When whole corn is stored in a cone-shaped pile 20 feet high, the diameter of the pile’s base is about 82 feet. (a) Find the angle of repose for whole corn. (b) Find when s 300 meters and s 1200 meters. (b) How tall is a pile of corn that has a base diameter of 100 feet? Model It 93. Photography A photographer is taking a picture of a three-foot-tall painting hung in an art gallery. The camera lens is 1 foot below the lower edge of the painting (see figure). The angle subtended by the camera lens x feet from the painting is arctan 3x x 2 4, x > 0. 3 ft 1 ft β θ α x Not drawn to scale 96. Angle of Elevation An airplane flies at an altitude of 6 miles toward a point directly over an observer. Consider and as shown in the figure. |
x θ x 6 mi Not drawn to scale (a) Write as a function of x. (b) Find when x 7 miles and x 1 mile. 97. Security Patrol A security car with its spotlight on is as parked 20 meters from a warehouse. Consider shown in the figure. and x (a) Use a graphing utility to graph as a function of x. θ 20 m (b) Move the cursor along the graph to approximate is maximum. the distance from the picture when (c) Identify the asymptote of the graph and discuss its meaning in the context of the problem. Not drawn to scale x (a) Write as a function of x. (b) Find when x 5 meters and x 12 meters. 333202_0407.qxd 12/7/05 11:10 AM Page 352 352 Chapter 4 Trigonometry Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 98–100, determine whether 98. sin 99. tan 5 6 5 4 1 2 1 arcsin 5 1 6 2 arctan 1 5 4 100. arctan x arcsin x arccos x 101. Define the inverse cotangent function by restricting the 0,, domain of the cotangent function to the interval and sketch its graph. 102. Define the inverse secant function by restricting the 0, 2 domain of the secant function to the intervals and sketch its graph. and 2,, 103. Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals 2, 0 and sketch its graph. 0, 2, and 107. Think About It Consider the functions given by f x sin x and f 1x arcsin x. functions (a) Use a graphing utility to graph the composite f 1 f. (b) Explain why the graphs in part (a) are not the graph and Why do the graphs of y x. f f 1 f f 1 and of the line f 1 f differ? (a) 108. Proof Prove each identity. arcsinx arcsin x arctanx arctan x, arctan x arctan (b) (c) (d) arcsin x arccos x (e) arcsin x arctan Skills Review 104. Use the results of Exercises 101–103 to evaluate each expression without using a calculator. (a) ( |
c) arcsec 2 arccot3 (b) arcsec 1 (d) arccsc 2 In Exercises 109–112, evaluate the expression. Round your result to three decimal places. 109. 111. 8.23.4 1.150 110. 112. 10142 162 105. Area In calculus, the y 1x 2 1, region bounded by and x a, it is shown that the area of y 0, the graphs of is given by x b Area arctan b arctan a In Exercises 113–116, sketch a right triangle corresponding to the trigonometric function of the acute angle Use the Pythagorean Theorem to determine the third side. Then find the other five trigonometric functions of.. (see figure). Find the area for the following values of and a 113. 115. sin 3 4 cos 5 6 114. 116. tan 2 sec 3 b. a 0, b 1 a 0, b 3 (a) (c) (b) (d) a 1, b 1 a 12 106. Think About It Use a graphing utility to graph the functions f x x and gx 6 arctan x. x > 0, it appears that For that there exists a positive real number a. for Approximate the number g > f. x > a. Explain why you know g < f such that a 117. Partnership Costs A group of people agree to share equally in the cost of a $250,000 endowment to a college. If they could find two more people to join the group, each person’s share of the cost would decrease by $6250. How many people are presently in the group? 118. Speed A boat travels at a speed of 18 miles per hour in still water. It travels 35 miles upstream and then returns to the starting point in a total of 4 hours. Find the speed of the current. 119. Compound Interest A total of $15,000 is invested in an account that pays an annual interest rate of 3.5%. Find the balance in the account after 10 years, if interest is compounded (a) quarterly, (b) monthly, (c) daily, and (d) continuously. 120. Profit Because of a slump in the economy, a department store finds that its annual profits have dropped from $742,000 in 2002 to $632,000 in 2004. The profit follows an exponential pattern of decline. What is the expected profit for 2008 |
? (Let represent 2002.) t 2 333202_0408.qxd 12/7/05 11:11 AM Page 353 4.8 Applications and Models Section 4.8 Applications and Models 353 What you should learn • Solve real-life problems involving right triangles. • Solve real-life problems involving directional bearings. • Solve real-life problems involving harmonic motion. Why you should learn it Right triangles often occur in real-life situations. For instance, in Exercise 62 on page 362, right triangles are used to determine the shortest grain elevator for a grain storage bin on a farm. Applications Involving Right Triangles In this section, the three angles of a right triangle are denoted by the letters and (where angles by the letters B,A, is the right angle), and the lengths of the sides opposite these is the hypotenuse). (where and b, a, C C c c Example 1 Solving a Right Triangle Solve the right triangle shown in Figure 4.78 for all unknown sides and angles. B a C c A 34.2° b = 19.4 FIGURE 4.78 it follows that A B 90 and B 90 34.2 55.8. Solution Because To solve for C 90, a, use the fact that tan A opp adj a b a b tan A. So, So, a 19.4 tan 34.2 13.18. cos A adj hyp c 19.4 23.46. b c cos 34.2 Similarly, to solve for use the fact that c b cos A c,. Now try Exercise 1. B Example 2 Finding a Side of a Right Triangle c = 110 ft A 72° C b A safety regulation states that the maximum angle of elevation for a rescue ladder A fire department’s longest ladder is 110 feet. What is the maximum safe is rescue height? 72. a Solution A sketch is shown in Figure 4.79. From the equation sin A ac, it follows that a c sin A 110 sin 72 104.6. So, the maximum safe rescue height is about 104.6 feet above the height of the fire truck. FIGURE 4.79 Now try Exercise 15. 333202_0408.qxd 12/7/05 11:11 AM Page 354 354 Chapter 4 Trigonometry Example 3 Finding a Side of a Right Triangle s a At a point 200 feet from the base of a building, the angle of elevation to the 53, bottom of a smokestack is |
as shown in Figure 4.80. Find the height of the smokestack alone. whereas the angle of elevation to the top is 35, s Solution Note from Figure 4.80 that this problem involves two right triangles. For the smaller right triangle, use the fact that tan 35 a 200 35° 53° 200 ft to conclude that the height of the building is a 200 tan 35. For the larger right triangle, use the equation FIGURE 4.80 tan 53 a s 200 a s 200 tan 53º. to conclude that s 200 tan 53 a 200 tan 53 200 tan 35 125.4 feet. Now try Exercise 19. So, the height of the smokestack is 20 m Angle of depression A FIGURE 4.81 Example 4 Finding an Acute Angle of a Right Triangle 1.3 m 2.7 m A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted so that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end, as shown in Figure 4.81. Find the angle of depression of the bottom of the pool. Solution Using the tangent function, you can see that tan A opp adj 2.7 20 0.135. So, the angle of depression is A arctan 0.135 0.13419 radian 7.69. Now try Exercise 25. 333202_0408.qxd 12/7/05 11:11 AM Page 355 Section 4.8 Applications and Models 355 Trigonometry and Bearings In surveying and navigation, directions are generally given in terms of bearings. A bearing measures the acute angle that a path or line of sight makes with a fixed E in north-south line, as shown in Figure 4.82. For instance, the bearing S Figure 4.82 means 35 degrees east of south. 35 N N 80° N 45° W E W E W E 35° S S 35° E S N 80° W S N 45° E FIGURE 4.82 Example 5 Finding Directions in Terms of Bearings A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 P.M. the ship changes course to N W, as shown in Figure 4.83. Find the ship’s bearing and distance from the port of departure at 3 P.M. 54 D b C FIGURE 4.83 20 nm d 54° B c Not |
drawn to scale E W N S 40 nm = 2(20 nm) A Solution For triangle can be determined to be b 20 sin 36 BCD, and d 20 cos 36. you have B 90 54 36. The two sides of this triangle For triangle ACD, tan A b you can find angle 20 sin 36 d 40 20 cos 36 40 A as follows. 0.2092494 A arctan 0.2092494 0.2062732 radian 11.82 90 11.82 78.18. you have Finally, from triangle The angle with the north-south line is the ship is yields N 78.18 W. ACD, So, the bearing of which sin A bc, c b sin A 20 sin 36 sin 11.82 57.4 nautical miles. Distance from port Now try Exercise 31. In air navigation, bearings are measured in degrees clockwise from north. Examples of air navigation bearings are shown below. 0° N 60° 270° W E 90° S 180° 0° N S 180° E 90° 225° 270° W 333202_0408.qxd 12/7/05 11:11 AM Page 356 356 Chapter 4 Trigonometry Harmonic Motion The periodic nature of the trigonometric functions is useful for describing the motion of a point on an object that vibrates, oscillates, rotates, or is moved by wave motion. For example, consider a ball that is bobbing up and down on the end of a spring, as shown in Figure 4.84. Suppose that 10 centimeters is the maximum distance the ball moves vertically upward or downward from its equilibrium (at rest) position. Suppose further that the time it takes for the ball to move from its maximum displacement above zero to its maximum displacement below zero and back again is seconds. Assuming the ideal conditions of perfect elasticity and no friction or air resistance, the ball would continue to move up and down in a uniform and regular manner. t 4 10 cm 10 cm 10 cm 0 cm 0 cm 0 cm −10 cm −10 cm −10 cm Equilibrium FIGURE 4.84 Maximum negative displacement Maximum positive displacement From this spring you can conclude that the period (time for one complete cycle) of the motion is Period 4 seconds its amplitude (maximum displacement from equilibrium) is Amplitude 10 centimeters and its frequency (number of cycles per second) is Frequency 1 4 cycle per second. Motion of this nature can be described by a sine or cosine function, and is |
called simple harmonic motion. 333202_0408.qxd 12/7/05 11:11 AM Page 357 Section 4.8 Applications and Models 357 Definition of Simple Harmonic Motion A point that moves on a coordinate line is said to be in simple harmonic motion if its distance from the origin at time d a sin t d a cos t is given by either or d t a where and a, period 2, are real numbers such that and frequency 2. > 0. The motion has amplitude Example 6 Simple Harmonic Motion Write the equation for the simple harmonic motion of the ball described in Figure 4.84, where the period is 4 seconds. What is the frequency of this harmonic motion? Solution Because the spring is at equilibrium d a sin t. d 0 when t 0, you use the equation Moreover, because the maximum displacement from zero is 10 and the period is 4, you have Amplitude a 10 Period 2 4. 2 Consequently, the equation of motion is d 10 sin t. 2 Note that the choice of initially moves up or down. The frequency is or a 10 a 10 depends on whether the ball Frequency 2 2 2 1 4 cycle per second. Now try Exercise 51. x One illustration of the relationship between sine waves and harmonic motion can be seen in the wave motion resulting when a stone is dropped into a calm pool of water. The waves move outward in roughly the shape of sine (or cosine) waves, as shown in Figure 4.85. As an example, suppose you are fishing and your fishing bob is attached so that it does not move horizontally. As the waves move outward from the dropped stone, your fishing bob will move up and down in simple harmonic motion, as shown in Figure 4.86. FIGURE 4.85 y FIGURE 4.86 333202_0408.qxd 12/7/05 11:11 AM Page 358 358 Chapter 4 Trigonometry Example 7 Simple Harmonic Motion Given the equation for simple harmonic motion d 6 cos 3 t 4 find (a) the maximum displacement, (b) the frequency, (c) the value of when t 4, t and (d) the least positive value of for which d 0. d Algebraic Solution The given equation has the form with 34. a 6 and d a cos t, Graphical Solution Use a graphing utility set in radian mode to graph a. The maximum displacement (from the point of equilibrium) is given by the amplitude. So, the maximum displacement is 6. |
y 6 cos 3 x. 4 a. Use the maximum feature of the graphing utility to estimate that y 0 the maximum displacement from the point of equilibrium is 6, as shown in Figure 4.87. b. c. 2 34 2 4 Frequency d 6 cos3 4 6 cos 3 61 6 3 8 cycle per unit of time y = 6 cos x3π ( ) 4 3 2 8 0 −8 FIGURE 4.87 d. To find the least positive value of for which t d 0, solve the equation 3 4 t 0. d 6 cos First divide each side by 6 to obtain cos 3 4 t 0. This equation is satisfied when Multiply these values by 43 to obtain t 2 3, 2, 10 3,.... 8 0 −8 So, the least positive value of t is t 2 3. FIGURE 4.88 Now try Exercise 55. b. The period is the time for the graph to complete one cycle, which You can estimate the frequency as follows. is x 2.667. Frequency 1 2.667 0.375 cycle per unit of time c. Use the trace feature to estimate that the value of when y is y 6, as shown in Figure 4.88. x 4 d. Use the zero or root feature to estimate that the least positive as shown in Figure is x 0.6667, for which y 0 x value of 4.89. y = 6 cos x3π ( ) 4 3 2 8 3 2 0 −8 FIGURE 4.89 333202_0408.qxd 12/7/05 11:11 AM Page 359 Section 4.8 Applications and Models 359 4.8 Exercises VOCABULARY CHECK: Fill in the blanks. 1. An angle that measures from the horizontal upward to an object is called the angle of ________, whereas an angle that measures from the horizontal downward to an object is called the angle of ________. 2. A ________ measures the acute angle a path or line of sight makes with a fixed north-south line. 3. A point that moves on a coordinate line is said to be in simple ________ ________ if its distance d a sin t from the origin at time is given by either d a cos t. or t d PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, solve the right triangle shown |
in the figure. Round your answers to two decimal places. 16. Length The sun is 20 above the horizon. Find the length of a shadow cast by a building that is 600 feet tall. b 10 b 24 b 10 c 52 1. 3. 5. 7. 9. 10. A 20, B 71, a 6, b 16, A 12 15, B 65 12, c 430.5 a 14.2 2. 4. 6. 8. B 54, A 8.4, a 25, b 1.32, c 15 a 40.5 c 35 c 9.45 17. Height A ladder 20 feet long leans against the side of a house. Find the height from the top of the ladder to the ground if the angle of elevation of the ladder is 80. 18. Height The length of a shadow of a tree is 125 feet when Approximate the 33. the angle of elevation of the sun is height of the tree FIGURE FOR 1–10 FIGURE FOR 11–14 In Exercises 11–14, find the altitude of the isosceles triangle shown in the figure. Round your answers to two decimal places. 11. 12. 13. 14. 52, 18, 41, 27, b 4 b 10 b 46 b 11 inches meters inches feet 15. Length The sun is 25 above the horizon. Find the length of a shadow cast by a silo that is 50 feet tall (see figure). 19. Height From a point 50 feet in front of a church, the angles of elevation to the base of the steeple and the top of 40, the steeple are (a) Draw right triangles that give a visual representation of the problem. Label the known and unknown quantities. respectively. and 47 35 (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) Find the height of the steeple. 20. Height You are standing 100 feet from the base of a platform from which people are bungee jumping. The angle of elevation from your position to the top of the platform from which they jump is From what height are the people jumping? 51. 21. Depth The sonar of a navy cruiser detects a submarine that is 4000 feet from the cruiser. The angle between the water line and the submarine is (see figure). How deep is the submarine? 34 34° 4000 ft 25° 50 ft Not drawn to scale 22. Angle of Elevation An engineer erects a 75-foot cellular telephone tower. Find |
the angle of elevation to the top of the tower at a point on level ground 50 feet from its base. 333202_0408.qxd 12/7/05 11:11 AM Page 360 360 Chapter 4 Trigonometry 121 2 23. Angle of Elevation The height of an outdoor basketball feet, and the backboard casts a shadow backboard is 171 feet long. 3 (a) Draw a right triangle that gives a visual representation of the problem. Label the known and unknown quantities. 30. Navigation A jet leaves Reno, Nevada and is headed The distance toward Miami, Florida at a bearing of between the two cities is approximately 2472 miles. 100. (a) How far north and how far west is Reno relative to Miami? (b) If the jet is to return directly to Reno from Miami, at (b) Use a trigonometric function to write an equation what bearing should it travel? involving the unknown quantity. (c) Find the angle of elevation of the sun. 24. Angle of Depression A Global Positioning System satellite orbits 12,500 miles above Earth’s surface (see figure). Find the angle of depression from the satellite to the horizon. Assume the radius of Earth is 4000 miles. 12,500 mi GPS satellite Angle of depression 4,000 mi Not drawn to scale 25. Angle of Depression A cellular telephone tower that is 150 feet tall is placed on top of a mountain that is 1200 feet above sea level. What is the angle of depression from the top of the tower to a cell phone user who is 5 horizontal miles away and 400 feet above sea level? 26. Airplane Ascent During takeoff, an airplane’s angle of ascent is 18 and its speed is 275 feet per second. (a) Find the plane’s altitude after 1 minute. (b) How long will it take the plane to climb to an altitude of 10,000 feet? 27. Mountain Descent A sign on a roadway at the top of a mountain indicates that for the next 4 miles the grade is 10.5 (see figure). Find the change in elevation over that distance for a car descending the mountain. Not drawn to scale 4 mi 10.5° 28. Mountain Descent A roadway sign at the top of a mountain indicates that for the next 4 miles the grade is 12%. Find the angle of the grade and the change in elevation over the 4 miles for a car descending the mountain. 29. Navigation An airplane flying at 600 miles per hour has a bearing of |
After flying for 1.5 hours, how far north and how far east will the plane have traveled from its point of departure? 52. 31. Navigation A ship leaves port at noon and has a bearing of S 29 W. The ship sails at 20 knots. (a) How many nautical miles south and how many nautical miles west will the ship have traveled by 6:00 P.M.? (b) At 6:00 P.M., the ship changes course to due west. Find the ship’s bearing and distance from the port of departure at 7:00 P.M. 32. Navigation A privately owned yacht leaves a dock in Myrtle Beach, South Carolina and heads toward Freeport The yacht averages in the Bahamas at a bearing of a speed of 20 knots over the 428 nautical-mile trip. S 1.4 E. (a) How long will it take the yacht to make the trip? (b) How far east and south is the yacht after 12 hours? (c) If a plane leaves Myrtle Beach to fly to Freeport, what bearing should be taken? 33. Surveying A surveyor wants to find the distance across a 32 is N W. C to swamp (see figure). The bearing from A, The surveyor walks 50 meters from the bearing to C is N W. Find (a) the bearing from B. A and at the point A B and (b) the distance from 68 to to B A B N S E W C 50 m A 34. Location of a Fire Two fire towers are 30 kilometers is due west of tower A fire is apart, where tower spotted from the towers, and the bearings from and are E N, respectively (see figure). Find the distance of the fire from the line segment N and W B. A 34 14 AB. B A d W N S E A 14° d 34° B 30 km Not drawn to scale 333202_0408.qxd 12/7/05 11:11 AM Page 361 35. Navigation A ship is 45 miles east and 30 miles south of port. The captain wants to sail directly to port. What bearing should be taken? 36. Navigation An airplane is 160 miles north and 85 miles east of an airport. The pilot wants to fly directly to the airport. What bearing should be taken? 37. Distance An observer in a lighthouse 350 feet above sea level observes two ships directly offshore. The angles of (see figure). How far depression to the ships are apart |
are the ships? 6.5 and 4 Section 4.8 Applications and Models 361 41. 42. L1: L2: L1: L2: 3x 2y 5 x y 1 2x y x 5y 8 4 43. Geometry Determine the angle between the diagonal of a cube and the diagonal of its base, as shown in the figure. 6.5° 4° 350 ft θ a a θ a a a FIGURE FOR 43 FIGURE FOR 44 Not drawn to scale cube and its edge, as shown in the figure. 44. Geometry Determine the angle between the diagonal of a 38. Distance A passenger in an airplane at an altitude of 10 kilometers sees two towns directly to the east of the plane. (see The angles of depression to the towns are figure). How far apart are the towns? and 28 55 45. Geometry Find the length of the sides of a regular penta- gon inscribed in a circle of radius 25 inches. 46. Geometry Find the length of the sides of a regular hexa- gon inscribed in a circle of radius 25 inches. 47. Hardware Write the distance across the flat sides of a hexagonal nut as a function of as shown in the figure. y r, 55° 28° 10 km Not drawn to scale 39. Altitude A plane is observed approaching your home and you assume that its speed is 550 miles per hour. The angle 16 of elevation of the plane is one at one time and minute later. Approximate the altitude of the plane. 57 40. Height While traveling across flat land, you notice a mountain directly in front of you. The angle of elevation to After you drive 17 miles closer to the the peak is 9. mountain, the angle of elevation is Approximate the height of the mountain. 2.5. r 60° x y 48. Bolt Holes The figure shows a circular piece of sheet metal that has a diameter of 40 centimeters and contains 12 equally spaced bolt holes. Determine the straight-line distance between the centers of consecutive bolt holes. 30° Geometry between two nonvertical lines satisfies the equation In Exercises 41 and 42, find the angle L1 The angle and L2. tan m2 1 m2 m1 m1 40 cm 35 cm m1 where and (Assume that m2 m1m2 are the slopes of 1. ) L1 and L2, respectively. 333202_0408.qxd 12/7/05 11 |
:11 AM Page 362 362 Chapter 4 Trigonometry Trusses unknown members of the truss. In Exercises 49 and 50, find the lengths of all the 49. 50. 35° 10 10 10 a b c 36 ft a 35° 10 6 ft 6 ft b 9 ft Harmonic Motion In Exercises 51–54, find a model for simple harmonic motion satisfying the specified conditions. Displacement t 0 51. 0 52. 0 53. 3 inches 54. 2 feet Amplitude Period 4 centimeters 3 meters 3 inches 2 feet 2 seconds 6 seconds 1.5 seconds 10 seconds Harmonic Motion In Exercises 55–58, for the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, t 5, and (d) the least positive (c) the value of when value of Use a graphing utility to verify your results. for which d 0. d t 55. 56. 57. 58. d 4 cos 8t d 1 2 cos 20t d 1 16 sin 120t d 1 64 sin 792t 59. Tuning Fork A point on the end of a tuning fork moves d a sin t. given that the tuning fork for middle C has a fre- in simple harmonic motion described by Find quency of 264 vibrations per second. 60. Wave Motion A buoy oscillates in simple harmonic motion as waves go past. It is noted that the buoy moves a total of 3.5 feet from its low point to its high point (see figure), and that it returns to its high point every 10 seconds. Write an equation that describes the motion of the buoy if its high point is at t 0. Equilibrium High point 3.5 ft Low point FIGURE FOR 60 61. Oscillation of a Spring A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by y 1 y is the time in seconds. 4 cos 16t t > 0, is measured in feet and where t (a) Graph the function. (b) What is the period of the oscillations? (c) Determine the first time the weight passes the point of equilibrium y 0. Model It 62. Numerical and Graphical Analysis A two-meterhigh fence is 3 meters from the side of a grain storage bin. A grain elevator must reach from ground level outside the fence to the storage |
bin (see figure). The objective is to determine the shortest elevator that meets the constraints. L2 θ L1 2 m θ 3 m (a) Complete four rows of the table. 0.1 0.2 L1 2 sin 0.1 2 sin 0.2 L2 L1 L2 3 cos 0.1 3 cos 0.2 23.0 13.1 333202_0408.qxd 12/7/05 11:11 AM Page 363 Model It (co n t i n u e d ) (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the minimum length of the elevator. L1 (c) Write the length L2 (d) Use a graphing utility to graph the function. Use the graph to estimate the minimum length. How does your estimate compare with that of part (b)? as a function of. 63. Numerical and Graphical Analysis The cross section of an irrigation canal is an isosceles trapezoid of which three of the sides are 8 feet long (see figure). The objective is to find the angle that maximizes the area of the cross. section. Hint: The area of a trapezoid is h2b1 b2 8 ft θ 8 ft θ Section 4.8 Applications and Models 363 (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. Graph the model with your scatter plot. How well does the model fit the data? (c) What is the period of the model? Do you think it is reasonable given the context? Explain your reasoning. (d) Interpret the meaning of the model’s amplitude in the context of the problem. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 65 and 66, determine whether 65. The Leaning Tower of Pisa is not vertical, but if you know the exact angle of elevation to the 191-foot tower when you stand near it, then you can determine the exact distance to the tower by using the formula d tan 191 d. 66. For the harmonic motion of a ball bobbing up and down on the end of a spring, one period can be described as the length of one coil of the spring. 8 ft 67. Writing Is it true that N 24 E means 24 degrees north of east? Explain. 68. Writing Explain the difference between bearings |
used in nautical navigation and bearings used in air navigation. Skills Review In Exercises 69 –72, write the slope-intercept form of the equation of the line with the specified characteristics.Then sketch the line. 69. m 4, passes through m 1 2, 71. Passes through passes through 70. 1, 2 1 3, 0 3, 2 and and 1 2, 1 3 2, 6 1 4, 2 3 72. Passes through (a) Complete seven additional rows of the table. Base 1 Base 2 Altitude Area 8 8 8 16 cos 10 8 16 cos 20 8 sin 10 8 sin 20 22.1 42.5 (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the maximum crosssectional area. (c) Write the area A as a function of. (d) Use a graphing utility to graph the function. Use the graph to estimate the maximum cross-sectional area. How does your estimate compare with that of part (b)? 64. Data Analysis The table shows the average sales (in millions of dollars) of an outerwear manufacturer for each t, month where represents January. t 1 S Time, t 1 2 3 4 5 6 Sales, s 13.46 11.15 8.00 4.85 2.54 1.70 Time, t 7 8 9 10 11 12 Sales, s 2.54 4.85 8.00 11.15 13.46 14.3 333202_040R.qxd 12/7/05 11:13 AM Page 364 364 Chapter 4 Trigonometry 4 Chapter Summary What did you learn? Section 4.1 Describe angles (p. 282). Use radian measure (p. 283). Use degree measure (p. 285). Use angles to model and solve real-life problems (p. 287). Section 4.2 Identify a unit circle and describe its relationship to real numbers (p. 294). Evaluate trigonometric functions using the unit circle (p. 295). Use domain and period to evaluate sine and cosine functions (p. 297). Use a calculator to evaluate trigonometric functions (p. 298). Section 4.3 Evaluate trigonometric functions of acute angles (p. 301). Use the fundamental trigonometric identities (p. 304). Use a calculator to evaluate trigonometric functions (p. 305). Use trigonometric functions to model and solve real-life problems (p. 306). |
Section 4.4 Evaluate trigonometric functions of any angle (p. 312). Use reference angles to evaluate trigonometric functions (p. 314). Evaluate trigonometric functions of real numbers (p. 315). Section 4.5 Use amplitude and period to help sketch the graphs of sine and cosine functions (p. 323). Sketch translations of the graphs of sine and cosine functions (p. 325). Use sine and cosine functions to model real-life data (p. 327). Section 4.6 Sketch the graphs of tangent (p. 332) and cotangent (p. 334) functions. Sketch the graphs of secant and cosecant functions (p. 335). Sketch the graphs of damped trigonometric functions (p. 337). Review Exercises 1, 2 3–6, 11–18 7–18 19–24 25–28 29–32 33–36 37–40 41–44 45–48 49–54 55, 56 57–70 71–82 83–88 89–92 93–96 97, 98 99–102 103–106 107, 108 Section 4.7 Evaluate and graph the inverse sine function (p. 343). Evaluate and graph the other inverse trigonometric functions (p. 345). Evaluate compositions of trigonometric functions (p. 347). 109–114, 123, 126 115–122, 124, 125 127–132 Section 4.8 Solve real-life problems involving right triangles (p. 353). Solve real-life problems involving directional bearings (p. 355). Solve real-life problems involving harmonic motion (p. 356). 133, 134 135 136 333202_040R.qxd 12/7/05 11:13 AM Page 365 4 Review Exercises In Exercises 1 and 2, estimate the angle to the nearest 4.1 one-half radian. 1. 2. In Exercises 3 –10, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle. 11 4 4 3 70 110 3. 5. 7. 9. 2 9 23 3 280 405 4. 6. 8. 10. Review Exercises 365 In Exercises 25–28, find the point 4.2 circle that corresponds to the real number x, y t. on the |
unit 25. 27. t 2 3 t 5 6 26. 28. t 3 4 t 4 3 In Exercises 29–32, evaluate (if possible) the six trigonometric functions of the real number. 29. 31. t 7 6 t 2 3 30. t 4 32. t 2 In Exercises 33–36, evaluate the trigonometric function using its period as an aid. In Exercises 11–14, convert the angle measure from degrees to radians. Round your answer to three decimal places. 11. 13. 480 33º 45 12. 14. 127.5 196 77 33. 35. sin 11 4 sin17 6 34. cos 4 36. cos13 3 In Exercises 15–18, convert the angle measure from radians to degrees. Round your answer to three decimal places. In Exercises 37–40, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. 5 7 3.5 15. 17. 16. 11 6 18. 5.7 37. tan 33 39. sec 12 5 38. 40. csc 10.5 sin 9 19. Arc Length Find the length of the arc on a circle with a 138. radius of 20 inches intercepted by a central angle of 20. Arc Length Find the length of the arc on a circle with a 60. radius of 11 meters intercepted by a central angle of 21. Phonograph Compact discs have all but replaced phonograph records. Phonograph records are vinyl discs that rotate on a turntable. A typical record album is 12 inches in 1 diameter and plays at 33 revolutions per minute. 3 (a) What is the angular speed of a record album? (b) What is the linear speed of the outer edge of a record album? 22. Bicycle At what speed is a bicyclist traveling when his 27-inch-diameter tires are rotating at an angular speed of 5 radians per second? 23. Circular Sector Find the area of the sector of a circle with a radius of 18 inches and central angle 120. 24. Circular Sector Find the area of the sector of a circle with a radius of 6.5 millimeters and central angle 56. In Exercises 41–44, find the exact values of the six 4.3 trigonometric functions of the angle shown in the figure. 41. 42. 4 θ 5 θ 6 43. 44. 6 8 θ 4 9 5 θ 333 |
202_040R.qxd 12/7/05 11:13 AM Page 366 366 Chapter 4 Trigonometry In Exercises 45– 48, use the given function value and trigonometric identities (including the cofunction identities) to find the indicated trigonometric functions. 45. sin 1 3 46. tan 4 47. csc 4 48. csc 5 (a) csc (c) sec (a) cot (c) cos (a) sin (c) sec (a) sin (c) tan (b) cos (d) tan (b) sec (d) csc (b) cos (d) tan (b) cot (d) sec 90 In Exercises 65–70, find the values of the six trigonometric functions of. Function Value sec 6 5 csc 3 2 sin 3 8 tan 5 4 cos 2 5 sin 2 4 65. 66. 67. 68. 69. 70. Constraint tan < 0 cos < 0 cos < 0 cos < 0 sin > 0 cos > 0 In Exercises 71–74, find the reference angle in standard position. and, and sketch In Exercises 49– 54, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. 49. 50. 51. 52. 53. 54. tan 33 csc 11 sin 34.2 sec 79.3 cot 15 14 cos 78 11 58 55. Railroad Grade A train travels 3.5 kilometers on a (see figure). What is straight track with a grade of the vertical rise of the train in that distance? 1 10 3.5 km 1°10′ Not drawn to scale 56. Guy Wire A guy wire runs from the ground to the top of a 25-foot telephone pole. The angle formed between the wire and the ground is How far from the base of the pole is the wire attached to the ground? 52. 71. 73. 264 6 5 72. 74. 635 17 3 In Exercises 75– 82, evaluate the sine, cosine, and tangent of the angle without using a calculator. 3 7 3 495 240 75. 77. 79. 81. 4 5 4 150 315 76. 78. 80. 82. In Exercises 83– 88, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. 83. 85. 85. sin 4 sin3.2 12 5 sin 84 |
. 86. 88. tan 3 cot4.8 tan25 7 In Exercises 89–96, sketch the graph of the function. 4.5 Include two full periods. In Exercises 57– 64, the point is on the terminal side in standard position. Determine the exact 4.4 of an angle values of the six trigonometric functions of the angle. 57. 58. 59. 60. 61. 62. 63. 64. 2 12, 16 3, 4 2 3, 5 10 3, 2 0.5, 4.5 0.3, 0.4 x, 4x, x > 0 2x, 3x, 3 x > 0 89. y sin x 91. 93. 95. f x 5 sin 2x 5 y 2 sin x gt 5 2 sint 90. 92. 94. 96. y cos x f x 8 cos x 4 y 4 cos x gt 3 cost 97. Sound Waves Sound waves can be modeled by sine y a sin bx, is measured in where x functions of the form seconds. (a) Write an equation of a sound wave whose amplitude is 2 and whose period is second. 1 264 (b) What is the frequency of the sound wave described in part (a)? 333202_040R.qxd 12/7/05 11:13 AM Page 367 98. Data Analysis: Meteorology The times S of sunset (Greenwich Mean Time) at 40 north latitude on the 15th of each month are: 1(16:59), 2(17:35), 3(18:06), 4(18:38), 5(19:08), 6(19:30), 7(19:28), 8(18:57), 9(18:09), 10(17:21), 11(16:44), 12(16:36). The month is represented by with corresponding to January. A model (in which minutes have been converted to the decimal parts of an hour) for the data is St 18.09 1.41 sint 4.60. 6 t 1 t, (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the model? Explain. In Exercises 99–106, sketch a graph of the function. |
4.6 Include two full periods. 99. f x tan x 100. f t tant 4 Review Exercises 367 In Exercises 119–122, use a calculator to evaluate the expression. Round your answer to two decimal places. 119. 121. arccos 0.324 tan11.5 120. 122. arccos0.888 tan1 8.2 In Exercises 123–126, use a graphing utility to graph the function. 123. f x 2 arcsin x 124. f x 3 arccos x 125. f x arctan x 2 126. f x arcsin 2x In Exercises 127–130, find the exact value of the expression. 127. 128. 129. 130. 4 cosarctan 3 tanarccos 3 5 secarctan 12 5 cotarcsin12 13 In Exercises 131 and 132, write an algebraic expression that is equivalent to the expression. 131. tanarccos secarcsinx 1 x 2 132. 101. 102. 103. 104. 105. 106. f x cot x gt 2 cot 2t f x sec x ht sect f x csc x f t 3 csc2t 4 4 4.8 133. Angle of Elevation The height of a radio transmission tower is 70 meters, and it casts a shadow of length 30 meters (see figure). Find the angle of elevation of the sun. In Exercises 107 and 108, use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of increases without bound. the function as x 107. f x x cos x 108. gx x4 cos x In Exercises 109–114, evaluate the expression. If 4.7 necessary, round your answer to two decimal places. 109. 111. 113. arcsin1 2 arcsin 0.4 sin10.44 110. 112. 114. arcsin1 arcsin 0.213 sin1 0.89 In Exercises 115–118, evaluate the expression without the aid of a calculator. 115. arccos 3 2 117. cos11 116. arccos 118. cos1 2 2 3 2 70 m θ 30 m 134. Height Your football has landed at the edge of the roof of your school building. When you are 25 feet from the base of the building, the angle of elevation to your football is How high |
off the ground is your football? 21. 135. Distance From city at a bearing of 810 miles at a bearing of A to city 48. C A B, a plane flies 650 miles to city B From city the plane flies 115. Find the distance from city to city C, and the bearing from city A to city C. 333202_040R.qxd 12/7/05 11:13 AM Page 368 368 Chapter 4 Trigonometry 136. Wave Motion Your fishing bobber oscillates in simple harmonic motion from the waves in the lake where you fish. Your bobber moves a total of 1.5 inches from its high point to its low point and returns to its high point every 3 seconds. Write an equation modeling the motion of your bobber if it is at its high point at time t 0. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 137–140, determine whether 137. The tangent function is often useful for modeling simple harmonic motion. 138. The inverse sine function y arcsin cannot be defined as a function over any interval that is greater than the 2 ≤ y ≤ 2. interval defined as y sin is not a function because sin 30 sin 150. 139. x 140. Because tan 34 1, arctan1 34. In Exercises 141–144, match the function with its graph. Base your selection solely on your interpretation a Explain your reasoning. [The of the constants graphs are labeled (a), (b), (c), and (d).] and y a sin bx b. (a) y (b) y x x π 2 −2 (c) y 3 2 1 −3 π 141. y 3 sin x 143. y 2 sin x 3 2 1 −d) 142. y 3 sin x 144. y 2 sin x 2 145. Writing Describe the behavior of g cos. zeros of Explain your reasoning. f sec at the (b) Make a conjecture about the relationship between tan 2 and cot. 147. Writing When graphing the sine and cosine functions, determining the amplitude is part of the analysis. Explain why this is not true for the other four trigonometric functions. 148. Oscillation of a Spring A weight is suspended from a ceiling by a steel spring. The weight is lifted (positive direction) from the equilibrium position and released. The resulting motion of the weight is modeled by y Aekt cos |
bt 1 5et10 cos 6t t y is the distance in feet from equilibrium and is where the time in seconds. The graph of the function is shown in the figure. For each of the following, describe the change in the system without graphing the resulting function. 1 3. 1 3. to is changed from 6 to 9. is changed from is changed from 1 5 1 10 (b) (a) (c) to A b k y 0.2 0.1 −0.1 −0.2 t 5π 149. Graphical Reasoning The formulas for the area of a A 1 s r, and is the angle measured circular sector and arc length are r respectively. ( in radians.) is the radius and 2 r2 (a) For 0.8, r. write the area and arc length as functions of What is the domain of each function? Use a graphing utility to graph the functions. Use the graphs to determine which function changes more r rapidly as r 10 centimeters, write the area and arc length as functions of What is the domain of each function? Use a graphing utility to graph and identify the functions. increases. Explain.. (b) For 146. Conjecture (a) Use a graphing utility to complete the table. 0.1 0.4 0.7 1.0 1.3 tan 2 cot 150. Writing Describe a real-life application that can be represented by a simple harmonic motion model and is different from any that you’ve seen in this chapter. Explain which function you would use to model your application and why. Explain how you would determine the amplitude, period, and frequency of the model for your application. 333202_040R.qxd 12/7/05 11:13 AM Page 369 Chapter Test 369 4 Chapter Test Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Consider an angle that measures 5 4 radians. (a) Sketch the angle in standard position. (b) Determine two coterminal angles (one positive and one negative). (c) Convert the angle to degree measure. x 2. A truck is moving at a rate of 90 kilometers per hour, and the diameter of its wheels is 1 meter. Find the angular speed of the wheels in radians per minute. 3. A water sprinkler sprays water on a lawn over a distance of |
25 feet and rotates through an angle of 130. Find the area of the lawn watered by the sprinkler. 4. Find the exact values of the six trigonometric functions of the angle shown in the figure. (−2, 6) y θ FIGURE FOR 4 5. Given that tan 3 2, 6. Determine the reference angle standard position. 7. Determine the quadrant in which 8. Find two exact values of a calculator.) find the other five trigonometric functions of. of the angle 290 and sketch and in sec < 0 lies if 0 ≤ < 360 and tan > 0. cos 32. if in degrees (Do not use 9. Use a calculator to approximate two values of Round the results to two decimal places. csc 1.030. in radians 0 ≤ < 2 if In Exercises 10 and 11, find the remaining five trigonometric functions of satisfying the conditions. cos 3 5, sec 17 8, tan < 0 sin > 0 11. 10. y 1 f − π −1 −2 In Exercises 12 and 13, sketch the graph of the function. (Include two full periods.) gx 2 sinx 12. 4 13. f 1 2 tan 2 x π π 2 In Exercises 14 and 15, use a graphing utility to graph the function. If the function is periodic, find its period. 14. y sin 2x 2 cos x 15. y 6e0.12t cos0.25t, 0 ≤ t ≤ 32 FIGURE FOR 16 and for the function c f x a sinbx c such that the graph of matches f 16. Find b,a, the figure. 17. Find the exact value of 18. Graph the function tanarccos 2 f x 2 arcsin 1 3 without the aid of a calculator. 2x. 19. A plane is 80 miles south and 95 miles east of Cleveland Hopkins International Airport. What bearing should be taken to fly directly to the airport? 20. Write the equation for the simple harmonic motion of a ball on a spring that starts at its lowest point of 6 inches below equilibrium, bounces to its maximum height of 6 inches above equilibrium, and returns to its lowest point in a total of 2 seconds. 333202_040R.qxd 12/7/05 11:13 AM Page 370 Proofs in Mathematics The Pythagorean Theorem The Pythagorean Theorem is one of the most famous theorems in mathematics |
. More than 100 different proofs now exist. James A. Garfield, the twentieth president of the United States, developed a proof of the Pythagorean Theorem in 1876. His proof, shown below, involved the fact that a trapezoid can be formed from two congruent right triangles and an isosceles right triangle. The Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse, where and are the legs and c a b is the hypotenuse. a2 b2 c2 a c b O c b Q a P Area of NOQ Proof N a M c b Area of MNOP trapezoid a ba b 1 2 Area of MNQ ab 1 2 1 2 Area of PQO ab 1 2 c2 1 2 a ba b ab 1 2 c2 a ba b 2ab c2 a2 2ab b2 2ab c2 a2 b2 c2 370 333202_040R.qxd 12/7/05 11:13 AM Page 371 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. The restaurant at the top of the Space Needle in Seattle, Washington is circular and has a radius of 47.25 feet. The dining part of the restaurant revolves, making about one complete revolution every 48 minutes. A dinner party was seated at the edge of the revolving restaurant at 6:45 P.M. and was finished at 8:57 P.M. (a) Find the angle through which the dinner party rotated. (b) Find the distance the party traveled during dinner. 2. A bicycle’s gear ratio is the number of times the freewheel turns for every one turn of the chainwheel (see figure). The table shows the numbers of teeth in the freewheel and chainwheel for the first five gears of an 18-speed touring bicycle. The chainwheel completes one rotation for each gear. Find the angle through which the freewheel turns for each gear. Give your answers in both degrees and radians. Gear number Number of teeth Number of teeth in freewheel in chainwheel 1 2 3 4 5 Freewheel 32 26 22 32 19 24 24 24 40 24 (a) What is the shortest distance d the helicopter would have to travel to land on the island? ( |
b) What is the horizontal distance that the helicopter would have to travel before it would be directly over the nearer end of the island? x (c) Find the width of the island. Explain how you obtained w your answer. 4. Use the figure below. F D E G B C A (a) Explain why triangles. ABC, ADE, and AFG are similar (b) What does similarity imply about the ratios BC AB, DE AD, and FG AF? A (c) Does the value of sin depend on which triangle from A part (a) is used to calculate it? Would the value of sin change if it were found using a different right triangle that was similar to the three given triangles? (d) Do your conclusions from part (c) apply to the other five trigonometric functions? Explain. 5. Use a graphing utility to graph h, and use the graph to decide Chainwheel (a) (b) 6. If f whether is even, odd, or neither. h hx cos2 x hx sin2 x is an even function and g is an odd function, use the 3. A surveyor in a helicopter is trying to determine the width of an island, as shown in the figure. 3000 ft 27° 39° d x w Not drawn to scale results of Exercise 5 to make a conjecture about where h, (a) hx f x2 hx gx2. 7. The model for the height (b) h (in feet) of a Ferris wheel car is h 50 50 sin 8t t where is the time (in minutes). (The Ferris wheel has a radius of 50 feet.) This model yields a height of 50 feet when t 0. Alter the model so that the height of the car is 1 foot when t 0. 371 333202_040R.qxd 12/7/05 11:13 AM Page 372 8. The pressure P (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by t P 100 20 cos8 3 where t is time (in seconds). (a) Use a graphing utility to graph the model. (b) What is the period of the model? What does the period tell you about this situation? (c) What is the amplitude of the model? What does it tell you about this situation? (d) If one cycle of this model is equivalent to one heart- beat, what is the pulse of this patient? |
(e) If a physician wants this patient’s pulse rate to be 64 beats per minute or less, what should the period be? What should the coefficient of be? t 9. A popular theory that attempts to explain the ups and downs of everyday life states that each of us has three cycles, called biorhythms, which begin at birth. These three cycles can be modeled by sine waves. Physical (23 days): P sin Emotional (28 days): E sin 2t 23, 2t 28, Intellectual (33 days): I sin 2t 33, where person who was born on July 20, 1986. is the number of days since birth. Consider a (a) Use a graphing utility to graph the three models in the same viewing window for 7300 ≤ t ≤ 7380. (b) Describe the person’s biorhythms during the month of September 2006. (c) Calculate the person’s three energy levels on September 22, 2006. 10. (a) Use a graphing utility to graph the functions given by f x 2 cos 2x 3 sin 3x and gx 2 cos 2x 3 sin 4x. (b) Use the graphs from part (a) to find the period of each function. (c) If and are positive integers, is the function given by hx A cos x B sin x periodic? Explain your reasoning. 372 11. Two trigonometric functions and have periods of 2, and f x 5.35. (a) Give one smaller and one larger positive value of at their graphs intersect at g x which the functions have the same value. (b) Determine one negative value of at which the graphs x intersect. (c) Is it true that reasoning. f 13.35 g4.65? Explain your 12. The function f t c f t. (a) f is periodic, with period Are the following equal? Explain. 2t 2c f 1 f t 1 (b) c. f t 2c f t f 1 2t t c f 1 2 (c) So, 13. If you stand in shallow water and look at an object below the surface of the water, the object will look farther away from you than it really is. This is because when light rays pass between air and water, the water refracts, or bends, the light rays. The index of refraction for water is 1.333. This (see figure |
). is the ratio of the sine of and the sine of 2 1 θ 2 θ 1 2 ft x d y (a) You are standing in water that is 2 feet deep and are (measured from a 2. looking at a rock at angle line perpendicular to the surface of the water). Find 60 1 (b) Find the distances and x y. (c) Find the distance d between where the rock is and where it appears to be. (d) What happens to d as you move closer to the rock? Explain your reasoning. 14. In calculus, it can be shown that the arctangent function can be approximated by the polynomial arctan x x x3 3 x7 7 x5 5 where x is in radians. (a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Study the pattern in the polynomial approximation of the arctangent function and guess the next term. Then repeat part (a). How does the accuracy of the approximation change when additional terms are added? 333202_0500.qxd 12/5/05 8:57 AM Page 373 Analytic Trigonometry 5.1 5.2 5.3 5.4 Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas 5.5 Multiple-Angle and Product-to-Sum Formula 55 Concepts of trigonometry can be used to model the height above ground of a seat on a Ferris wheel AT I O N S Trigonometric equations and identities have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Friction, • Data Analysis: Unemployment Rate, • Projectile Motion, Exercise 99, page 381 Exercise 76, page 398 Exercise 101, page 421 • Shadow Length, Exercise 56, page 388 • Ferris Wheel, Exercise 75, page 398 • Harmonic Motion, Exercise 75, page 405 • Mach Number, Exercise 121, page 417 • Ocean Depth, Exercise 10, page 428 373 333202_0501.qxd 12/5/05 9:15 AM Page 374 374 Chapter 5 Analytic Trigonometry 5.1 Using Fundamental Identities What you should learn • Recognize and write the fundamental trigonometric identities. • Use the fundamental trigonometric identities to evaluate trigon |
ometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions. Why you should learn it Fundamental trigonometric identities can be used to simplify trigonometric expressions. For instance, in Exercise 99 on page 381, you can use trigonometric identities to simplify an expression for the coefficient of friction. Introduction In Chapter 4, you studied the basic definitions, properties, graphs, and applications of the individual trigonometric functions. In this chapter, you will learn how to use the fundamental identities to do the following. 1. Evaluate trigonometric functions. 2. Simplify trigonometric expressions. 3. Develop additional trigonometric identities. 4. Solve trigonometric equations. Fundamental Trigonometric Identities Reciprocal Identities sin u 1 csc u csc u 1 sin u cos u 1 sec u sec u 1 cos u tan u 1 cot u cot u 1 tan u Quotient Identities tan u sin u cos u Pythagorean Identities cot u cos u sin u sin2 u cos2 u 1 1 tan2 u sec2 u 1 cot 2 u csc2 u Cofunction Identities sin 2 tan 2 sec 2 u cos u u cot u u csc u Even/Odd Identities cos 2 cot 2 csc 2 u sin u u tan u u sec u sinu sin u cscu csc u cosu cos u secu sec u tanu tan u cotu cot u Pythagorean identities are sometimes used in radical form such as sin u ± 1 cos2 u The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. or tan u ± sec 2 u 1 where the sign depends on the choice of u. 333202_0501.qxd 12/5/05 9:15 AM Page 375 Section 5.1 Using Fundamental Identities 375 Using the Fundamental Identities One common use of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions. Example 1 Using Identities to Evaluate a Function Use the values trigonometric functions. sec u 3 2 and tan u > 0 to find the values of all six You should learn the fundamental trigonometric identities well, because they are used frequently in trigonometry and they will also appear later in calculus. Note that u can be an angle, a real number, or a variable. Solution |
Using a reciprocal identity, you have 2 3 cos u 1 sec u 1 32. Using a Pythagorean identity, you have sin2 u 1 cos. Pythagorean identity Substitute 2 3 for cos u. Simplify. sec u < 0 Because Moreover, because sin u 53. the negative root and obtain sine and cosine, you can find the values of all six trigonometric functions. lies in Quadrant III. is in Quadrant III, you can choose Now, knowing the values of the tan u > 0, is negative when it follows that and sin u u u sin u 5 3 cos u 2 3 tan u sin u cos u csc u 1 sin u sec u 1 cos u 3 5 35 5 3 2 53 23 5 2 cot u 1 tan u 2 5 25 5 Now try Exercise 11. Te c h n o l o g y You can use a graphing utility to check the result of Example 2. To do this, graph sin x cos 2 x sin x y1 and sin3 x y2 in the same viewing window, as shown below. Because Example 2 shows the equivalence algebraically and the two graphs appear to coincide, you can conclude that the expressions are equivalent. −π 2 −2 Example 2 Simplifying a Trigonometric Expression π Simplify sin x cos 2 x sin x. Solution First factor out a common monomial factor and then use a fundamental identity. sin x cos 2 x sin x sin xcos2 x 1 Factor out common monomial factor. sin x1 cos 2 x sin xsin2 x sin3 x Factor out 1. Pythagorean identity Multiply. Now try Exercise 45. 333202_0501.qxd 12/5/05 9:15 AM Page 376 376 Chapter 5 Analytic Trigonometry When factoring trigonometric expressions, it is helpful to find a special polynomial factoring form that fits the expression, as shown in Example 3. Example 3 Factoring Trigonometric Expressions Factor each expression. a. sec2 1 b. 4 tan2 tan 3 Solution a. Here you have the difference of two squares, which factors as sec2 1 sec 1sec 1). b. This expression has the polynomial form ax 2 bx c, and it factors as 4 tan2 tan 3 4 tan 3tan 1. Now try Exercise 47. On occasion, factoring or simplifying can best be done by first rewriting the expression in terms of just one trigon |
ometric function or in terms of sine and cosine only. These strategies are illustrated in Examples 4 and 5, respectively. Example 4 Factoring a Trigonometric Expression Factor csc2 x cot x 3. Solution Use the identity cotangent. csc2 x 1 cot 2 x to rewrite the expression in terms of the csc2 x cot x 3 1 cot 2 x cot x 3 cot 2 x cot x 2 cot x 2cot x 1 Now try Exercise 51. Pythagorean identity Combine like terms. Factor. Example 5 Simplifying a Trigonometric Expression Simplify sin t cot t cos t. Remember that when adding rational expressions, you must first find the least common denominator (LCD). In Example 5, the LCD is sin t. Solution Begin by rewriting cot in terms of sine and cosine. t sin t cot t cos t sin t cos t sin t sin2 t cos 2 t sin t cos t 1 sin t csc t Now try Exercise 57. Quotient identity Add fractions. Pythagorean identity Reciprocal identity 333202_0501.qxd 12/5/05 9:15 AM Page 377 Section 5.1 Using Fundamental Identities 377 Example 6 Adding Trigonometric Expressions Perform the addition and simplify. sin 1 cos cos sin Solution sin 1 cos cos sin sin sin (cos 1 cos 1 cos sin sin2 cos2 cos 1 cos sin 1 cos 1 cos sin Multiply. Pythagorean identity: sin2 cos2 1 1 sin csc Now try Exercise 61. Divide out common factor. Reciprocal identity The last two examples in this section involve techniques for rewriting expres- sions in forms that are used in calculus. Example 7 Rewriting a Trigonometric Expression Rewrite 1 1 sin x so that it is not in fractional form. Solution cos 2 x 1 sin2 x 1 sin x1 sin x, From the Pythagorean identity you can see that multiplying both the numerator and the denominator by 1 sin x will produce a monomial denominator. 1 1 sin x 1 sin x 1 sin x 1 1 sin x 1 sin x 1 sin2 x 1 sin x cos 2 x 1 cos2 x sin x cos2 x 1 cos 2 x sin x cos x 1 cos x sec2 x tan x sec x Now try Exercise 65. Multiply numerator and denominator by 1 sin x. Multiply. Pyth |
agorean identity Write as separate fractions. Product of fractions Reciprocal and quotient identities 333202_0501.qxd 12/5/05 9:15 AM Page 378 378 Chapter 5 Analytic Trigonometry Example 8 Trigonometric Substitution Use the substitution 4 x 2 x 2 tan, 0 < < 2, to write as a trigonometric function of. Solution Begin by letting x 2 tan. Then, you can obtain 4 x 2 4 2 tan 2 4 4 tan2 41 tan2 4 sec2 2 sec. Now try Exercise 77. Substitute 2 tan for x. Rule of exponents Factor. Pythagorean identity sec > 0 for 0 < < 2 4 + x2 x θ = arctan x 2 2 Angle whose tangent is x 2. FIGURE 5.1 Figure 5.1 shows the right triangle illustration of the trigonometric substitux 2 tan tion in Example 8. You can use this triangle to check the solution of 0 < < 2, Example 8. For adj 2, opp x, hyp 4 x 2. you have and With these expressions, you can write the following. sec hyp adj 4 x 2 2 sec 2 sec 4 x 2 So, the solution checks. Example 9 Rewriting a Logarithmic Expression Rewrite lncsc lntan as a single logarithm and simplify the result. Solution lncsc lntan ln csc tan sin cos ln 1 ln 1 cos sin lnsec Now try Exercise 91. Product Property of Logarithms Reciprocal and quotient identities Simplify. Reciprocal identity 333202_0501.qxd 12/5/05 9:15 AM Page 379 5.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. Section 5.1 Using Fundamental Identities 379 VOCABULARY CHECK: Fill in the blank to complete the trigonometric identity. sin u cos u ________ ________ 1 tan u 1 ________ csc2 u u ________ sin 2 cosu ________ 1. 3. 5. 7. 9. 1 sec u ________ ________ 1 sin u 1 tan2 u ________ u ________ sec 2 tanu ________ 2. 4. 6. 8. 10. PREREQUISITE SKILLS REVIEW |
: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions. 19. sinx cosx 20. sin2 x cos2 x 1. sin x 2. tan x 3 2 3 3,, 3. sec 2, cos x cos x 1 2 3 2 2 2 sin 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. csc 5 3, tan x 5 12, cot 3, sin tan 3 4 sec x 13 12 10 10 csc 35 5 cos x 4 5 2 4 tan x,,, sec 3 2 x 3 5 cos 2 sinx 1 3 sec x 4, tan 2, csc 5, sin 1, tan sin x > 0 sin < 0 cos < 0 cot 0 is undefined, sin > 0 In Exercises 15–20, match the trigonometric expression with one of the following. (a) sec x (d) 1 (b) (e) 1 tan x (c) cot x (f) sin x 15. 17. sec x cos x cot2 x csc 2 x 16. 18. tan x csc x 1 cos2 xcsc x In Exercises 21–26, match the trigonometric expression with one of the following. (b) tan x sec2 x (a) csc x (d) sin x x tan (e) 21. 23. 25. sin x sec x sec4 x tan4 x sec2 x 1 sin2 x (c) (f) sin2 x sec2 x tan2 x 22. 24. 26. cos2 xsec2 x 1 cot x sec x cos22 x cos x In Exercises 27–44, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. 27. 29. 31. 33. 35. cot sec sin csc sin cot x csc x 1 sin2 x csc2 x 1 sec sin tan xsec x 39. 37. cos 2 cos2 y 1 sin y sin tan cos cot u sin u tan u cos u 43. 44. sin sec cos csc 41. 28. 30. 32. 34. 36. 38. cos tan sec2 x1 sin2 x csc sec 1 tan |
2 x 1 tan2 sec2 cot 2 xcos x 40. cos t1 tan2 t 42. csc tan sec 333202_0501.qxd 12/5/05 9:15 AM Page 380 380 Chapter 5 Analytic Trigonometry In Exercises 45–56, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. 71. 72. y1 y1 cos x 1 sin x, y2 sec4 x sec2 x, 1 sin x cos x tan2 x tan4 x y2 45. 47. 49. 51. 53. 55. 56. tan2 x tan2 x sin2 x sin2 x sec2 x sin2 x sec2 x 1 sec x 1 tan4 x 2 tan2 x 1 sin4 x cos4 x csc3 x csc2 x csc x 1 sec3 x sec2 x sec x 1 46. 48. 50. 52. 54. sin2 x csc2 x sin2 x cos2 x cos2 x tan2 x cos2 x 4 cos x 2 1 2 cos2 x cos4 x sec4 x tan4 x In Exercises 57– 60, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer. 57. 58. 59. 60. sin x cos x2 cot x csc xcot x csc x 2 csc x 22 csc x 2 3 3 sin x3 3 sin x In Exercises 61–64, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer. 61. 63. 1 1 cos x cos x 1 sin x 1 1 cos x 1 sin x cos x 62. 64. 1 sec x 1 1 sec x 1 tan x sec2 x tan x In Exercises 65– 68, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer. 65. 67. sin2 y 1 cos y 3 sec x tan x 66. 68. 5 tan x sec x tan2 x csc x 1 Numerical and Graphical Analysis In Exercises 69 –72, use a graphing utility to complete the table and graph the functions. Make a conjecture about and y1 y2. In Exercises 73–76, use a graphing utility to determine which of the six trig |
onometric functions is equal to the expression. Verify your answer algebraically. 73. 74. 75. 76. cos x cot x sin x sec x csc x tan x cos x 1 1 cos x sin x 1 sin cos 1 sin cos 1 2 In Exercises 77– 82, use the trigonometric substitution to write the algebraic expression as a trigonometric function of where, 0 < < /2. x 3 cos 77. 78. 79. 80. 81. 82. 9 x 2, 64 16x 2, x 2 9, x 2 4, x 2 25, x 2 100, x 2 cos x 3 sec x 2 sec x 5 tan x 10 tan In Exercises 83– 86, use the trigonometric substitution to write the algebraic equation as a trigonometric function of where Then find sin and cos., /2 < < /2. x 3 sin x 6 sin 3 9 x2, 3 36 x2, 22 16 4x2, 53 100 x2, 83. 84. 85. 86. x 2 cos x 10 cos In Exercises 87–90, use a graphing utility to solve the 0 ≤ < 2. equation for where, 87. 88. 89. 90. sin 1 cos2 cos 1 sin2 sec 1 tan2 csc 1 cot2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 In Exercises 91–94, rewrite the expression as a single logarithm and simplify the result. x y1 y2 cos x, 2 sec x cos x, 69. 70. y1 y1 sin x y2 y2 sin x tan x 91. 92. lncos x lnsin x lnsec x lnsin x lncot t ln1 tan2 t 93. 94. lncos2 t ln1 tan2 t 333202_0501.qxd 12/5/05 9:15 AM Page 381 In Exercises 95–98, use a calculator to demonstrate the identity for each value of. 95. csc2 cot2 1 (a) 132, (b) 2 7 96. 97. 98. tan2 1 sec2 346, (a) (b) cos sin 2 80, (a) (b) sin sin 250, (a) (b) 3.1 0.8 1 2 99. Friction The forces |
acting on an object weighing W units on an inclined plane positioned at an angle of with the horizontal (see figure) are modeled by W cos W sin where for and simplify the result. is the coefficient of friction. Solve the equation W θ 100. Rate of Change The rate of change of the function f x csc x sin x is given by the expression csc x cot x cos x. Show that this expression can also be written as cos x cot2 x. Synthesis True or False? In Exercises 101 and 102, determine whether the statement is true or false. Justify your answer. 101. The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative. 102. A cofunction identity can be used to transform a tangent function so that it can be represented by a cosecant function. Section 5.1 Using Fundamental Identities 381 In Exercises 103–106, fill in the blanks. (Note: The notation x → c from the right and x → c indicates that indicates that approaches x approaches x c 103. As 104. As 105. As 106. As 2, sin x → x → x → 0, cos x →, tan x → x → x →, sin x → 2 from the left.) c csc x →. sec x →. cot x →. csc x →. and and and and In Exercises 107–112, determine whether or not the equation is an identity, and give a reason for your answer. 108. cot csc2 1 is a constant. 107. 109. 110. 111. k tan, cos 1 sin2 sin k cos k 1 5 cos sin csc 1 5 sec 112. csc2 1 113. Use the definitions of sine and cosine to derive the Pythagorean identity sin2 cos2 1. 114. Writing Use the Pythagorean identity sin2 cos2 1 identities, to derive the 1 tan2 sec2 Discuss how to remember these identities and other fundamental identities. Pythagorean 1 cot2 csc2. other and Skills Review In Exercises 115 and 116, perform the operation and simplify. 115. x 5x 5 116. 2z 32 In Exercises 117–120, perform the addition or subtraction and simplify. 117. 119. 1 x 5 2x x2 4 x x 8 7 x 4 118. 120. 6x |
x 4 x x2 25 3 4 x x2 x 5 In Exercises 121–124, sketch the graph of the function. (Include two full periods.) 121. 123. f x 1 2 f x 1 2 sin x 122. f x 2 tan x 2 secx 4 124. f x 3 2 cosx 3 333202_0502.qxd 12/5/05 9:01 AM Page 382 382 Chapter 5 Analytic Trigonometry 5.2 Verifying Trigonometric Identities What you should learn • Verify trigonometric identities. Why you should learn it You can use trigonometric identities to rewrite trigonometric equations that model real-life situations. For instance, in Exercise 56 on page 388, you can use trigonometric identities to simplify the equation that models the length of a shadow cast by a gnomon (a device used to tell time). Introduction In this section, you will study techniques for verifying trigonometric identities. In the next section, you will study techniques for solving trigonometric equations. The key to verifying identities and solving equations is the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions. Remember that a conditional equation is an equation that is true for only some of the values in its domain. For example, the conditional equation sin x 0 Conditional equation x n, is true only for are solving the equation. where n is an integer. When you find these values, you On the other hand, an equation that is true for all real values in the domain of the variable is an identity. For example, the familiar equation sin2 x 1 cos2 x Identity is true for all real numbers So, it is an identity. x. Verifying Trigonometric Identities Although there are similarities, verifying that a trigonometric equation is an identity is quite different from solving an equation. There is no well-defined set of rules to follow in verifying trigonometric identities, and the process is best learned by practice. Guidelines for Verifying Trigonometric Identities 1. Work with one side of the equation at a time. It is often better to work with the more complicated side first. Robert Ginn/PhotoEdit 2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, |
as do secants and tangents, and cosecants and cotangents. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Always try something. Even paths that lead to dead ends provide insights. Verifying trigonometric identities is a useful process if you need to convert a trigonometric expression into a form that is more useful algebraically. When you verify an identity, you cannot assume that the two sides of the equation are equal because you are trying to verify that they are equal. As a result, when verifying identities, you cannot use operations such as adding the same quantity to each side of the equation or cross multiplication. 333202_0502.qxd 12/5/05 9:01 AM Page 383 Section 5.2 Verifying Trigonometric Identities 383 Example 1 Verifying a Trigonometric Identity Verify the identity sec2 1 sec2 sin2. Remember that an identity is only true for all real values in the domain of the variable. For instance, in Example 1 the identity is not true when 2 is because not defined when 2. sec2 Solution Because the left side is more complicated, start with it. tan2 1 1 sec2 sec2 1 sec2 Pythagorean identity tan2 sec2 tan2 cos 2 sin2 cos2 cos2 sin2 Simplify. Reciprocal identity Quotient identity Simplify. Notice how the identity is verified. You start with the left side of the equation (the more complicated side) and use the fundamental trigonometric identities to simplify it until you obtain the right side. Now try Exercise 5. There is more than one way to verify an identity. Here is another way to verify the identity in Example 1. sec2 1 sec2 sec2 1 sec2 sec2 1 cos2 sin2 Rewrite as the difference of fractions. Reciprocal identity Pythagorean identity Example 2 Combining Fractions Before Using Identities Verify the identity 1 1 sin 1 1 sin 2 sec2. Solution 1 1 sin 1 1 sin 1 sin 1 sin 1 sin 1 sin 2 1 sin2 2 cos2 2 sec2 Now try Exercise 19. Add fractions. Simplify. Pythagorean identity Reciprocal identity 333202_0502.qxd 12/5/05 9:01 AM Page 384 384 Chapter 5 Analytic Trigonometry Example 3 Verifying Trigonometric Identity Verify the identity tan2 x 1cos2 x 1 tan2 x. Al |
gebraic Solution By applying identities before multiplying, you obtain the following. tan2 x 1cos2 x 1 sec2 xsin2 x Pythagorean identities sin2 x cos 2 x sin x cos x tan2 x 2 Reciprocal identity Rule of exponents Quotient identity Numerical Solution Use the table feature of a graphing utility set in radian mode to create a table that shows the tan2 x 1cos2 x 1 y1 and values of tan2 x y2 for different values of as shown in Figure 5.2. From the table you can see that the y2 values of appear to be identical, so tan2 x 1cos2 x 1 tan2 x appears to be an identity. and y1 x, Now try Exercise 39. FIGURE 5.2 Example 4 Converting to Sines and Cosines Verify the identity tan x cot x sec x csc x. Solution Try converting the left side into sines and cosines. Although a graphing utility can be useful in helping to verify an identity, you must use algebraic techniques to produce a valid proof. tan x cot x sin x cos x cos x sin x sin2 x cos 2 x cos x sin x Quotient identities Add fractions. Pythagorean identity 1 cos x sin x 1 cos x 1 sin x sec x csc x Reciprocal identities As shown at the right, csc2 x1 cos x is considered a 11 cos x simplified form of because the expression does not contain any fractions. Now try Exercise 29. Recall from algebra that rationalizing the denominator using conjugates is, on occasion, a powerful simplification technique. A related form of this technique, shown below, works for simplifying trigonometric expressions as well. 1 cos x 1 cos x 1 cos x sin2 x 1 cos x 1 cos2 x 1 1 cos x 1 1 cos x csc2 x1 cos x This technique is demonstrated in the next example. 333202_0502.qxd 12/5/05 9:01 AM Page 385 Section 5.2 Verifying Trigonometric Identities 385 Example 5 Verifying Trigonometric Identities Verify the identity sec y tan y cos y 1 sin y. Solution Begin with the right side, because you can create a monomial denominator by multiplying the numerator and denominator by 1 sin y. cos y 1 sin y 1 sin y cos y 1 sin y 1 sin y cos y cos y sin y |
1 sin2 y cos y cos y sin y cos 2 y cos y sin y cos2 y cos y cos2 y 1 cos y sin y cos y sec y tan y Now try Exercise 33. Multiply numerator and denominator by 1 sin y. Multiply. Pythagorean identity Write as separate fractions. Simplify. Identities In Examples 1 through 5, you have been verifying trigonometric identities by working with one side of the equation and converting to the form given on the other side. On occasion, it is practical to work with each side separately, to obtain one common form equivalent to both sides. This is illustrated in Example 6. Example 6 Working with Each Side Separately Verify the identity cot 2 1 csc 1 sin sin. Solution Working with the left side, you have cot 2 1 csc csc2 1 1 csc csc 1csc 1 1 csc csc 1. Now, simplifying the right side, you have 1 sin sin 1 sin sin sin csc 1. Pythagorean identity Factor. Simplify. Write as separate fractions. Reciprocal identity The identity is verified because both sides are equal to csc 1. Now try Exercise 47. 333202_0502.qxd 12/5/05 9:01 AM Page 386 386 Chapter 5 Analytic Trigonometry In Example 7, powers of trigonometric functions are rewritten as more complicated sums of products of trigonometric functions. This is a common procedure used in calculus. Example 7 Three Examples from Calculus Verify each identity. a. b. c. tan4 x tan2 x sec2 x tan2 x sin3 x cos4 x cos4 x cos 6 x sin x csc4 x cot x csc2 xcot x cot3 x Solution a. tan4 x tan2 xtan2 x tan2 xsec2 x 1 tan2 x sec2 x tan2 x sin3 x cos4 x sin2 x cos4 x sin x b. 1 cos2 xcos4 x sin x cos4 x cos6 x sin x c. csc4 x cot x csc2 x csc2 x cot x csc2 x1 cot2 x cot x csc2 xcot x cot3 x Now try Exercise 49. Write as separate factors. Pythagorean identity Multiply. Write as separate factors. Pythagorean identity Multiply. Write as separate factors. Pyth |
agorean identity Multiply. W RITING ABOUT MATHEMATICS Error Analysis You are tutoring a student in trigonometry. One of the homework problems your student encounters asks whether the following statement is an identity. tan2 x sin2 x? 5 6 tan2 x Your student does not attempt to verify the equivalence algebraically, but mistakenly uses only a graphical approach. Using range settings of Xmin 3 Xmax 3 Xscl 2 Ymin 20 Ymax 20 Yscl 1 your student graphs both sides of the expression on a graphing utility and concludes that the statement is an identity. What is wrong with your student’s reasoning? Explain. Discuss the limitations of verifying identities graphically. 333202_0502.qxd 12/5/05 9:01 AM Page 387 Section 5.2 Verifying Trigonometric Identities 387 5.2 Exercises VOCABULARY CHECK: In Exercises 1 and 2, fill in the blanks. 1. An equation that is true for all real values in its domain is called an ________. 2. An equation that is true for only some values in its domain is called a ________ ________. In Exercises 3–8, fill in the blank to complete the trigonometric identity. 3. 1 cot u ________ 5. sin2 u ________ 1 7. cscu ________ 4. 6. 8. ________ cos u sin u cos 2 secu ________ u ________ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1 csc x 1 1 1 sin x 1 cos x cos x sin x cos x sin x cos x 1 tan x tan 1 26. cot x tan 2 cscx secx 1 sin y1 siny cos2 y tan x cot x cos x sec x cos2 x sin2 x tan x In Exercises 1–38, verify the identity. 1. 3. 4. 5. 6. 7. 8. 9. 11. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 2. sec y cos y 1 sin t csc t 1 1 sin 1 sin cos 2 cot 2 ysec 2 y 1 1 cos2 sin2 1 2 sin2 cos2 sin2 2 cos 2 1 sin2 sin4 cos2 cos4 cos x sin x |
tan x sec x csc2 cot cot2 t csc t sin12 x cos x sin52 x cos x cos3 xsin x sec6 xsec x tan x sec4 xsec x tan x sec5 x tan3 x csc t sin t csc sec 1 tan cot3 t csc t 12. 10. cos t csc2 t 1 tan sec2 tan csc x sin x 1 sec x tan x sec 1 1 cos csc x sin x cos x cot x sec x cos x sin x tan x sec 1 tan x 1 cot x 1 csc x 1 sin x cos cot 1 sin 1 sin cos tan x cot x csc x sin x 1 csc cos 1 sin 2 sec 23. 24. 25. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. sin x sin y cos x cos y 0 cot x cot y cot x cot y 1 tan y cot x tan x tan y 1 tan x tan y tan x cot y tan x cot y cos x cos y sin x sin y 1 sin 1 sin 1 cos 1 cos cos2 cos2 2 sec2 y cot 2 2 sin t csc 2 x 1 cot2 x 1 sin cos 1 cos sin 1 t tan t y 1 38. sec2 2 333202_0502.qxd 12/5/05 9:01 AM Page 388 388 Chapter 5 Analytic Trigonometry In Exercises 39– 46, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically. 39. 40. 41. 42. 43. 44. 45. 2 sec2 x 2 sec2 x sin2 x sin2 x cos 2 x 1 csc xcsc x sin x sin x cos x cot x csc2 x sin x 2 cos 2 x 3 cos4 x sin2 x3 2 cos2 x tan 4 x tan2 x 3 sec2 x4 tan2 x 3 csc4 x 2 csc2 x 1 cot4 x sin4 2 sin2 1 cos cos5 cot csc 1 1 sin x cos x cos x 1 sin x 46. csc |
1 cot 47. 48. In Exercises 47–50, verify the identity. tan5 x tan3 x sec2 x tan3 x sec4 x tan2 x tan2 x tan4 x sec2 x cos3 x sin2 x sin2 x sin4 x cos x sin4 x cos4 x 1 2 cos2 x 2 cos4 x 49. 50. In Exercises 51–54, use the cofunction identities to evaluate the expression without the aid of a calculator. 51. 53. 54. sin2 25 sin2 65 cos2 20 cos2 52 cos2 38 cos2 70 sin2 12 sin2 40 sin2 50 sin2 78 52. cos2 55 cos2 35 55. Rate of Change The rate of change of the function f x sin x csc x with respect to change in the variable x is given by the expression Show that the expression for the rate of change can also be cos x cot2 x. cos x csc x cot x. Model It s 56. Shadow Length The length of a shadow cast by a h (see vertical gnomon (a device used to tell time) of height when the angle of the sun above the horizon is figure) can be modeled by the equation s h sin90. sin h ft θ s Model It (co n t i n u e d ) (a) Verify that the equation for s is equal to h cot. (b) Use a graphing utility to complete the table. Let h 5 feet. 10 20 30 40 50 60 70 80 90 s s (c) Use your table from part (b) to determine the angles of the sun for which the length of the shadow is the greatest and the least. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is 90? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 57 and 58, determine whether sin2 cos2 1 tan2 is an identity, 57. The equation because sin20 cos20 1 and 1 tan2 1 cot2 1 tan20 1. 58. The equation is not 1 tan26 11 3, an and identity, because it is true that 1 cot26 4. Think About It In Exercises 59 and 60, explain why the equation is not an identity and find one value of the variable for which the equation is not true |
. 59. 60. sin 1 cos2 tan sec2 1 Skills Review In Exercises 61–64, perform the operation and simplify. 61. 63. 2 3i 26 16 1 4 62. 64. 2 5i 2 3 2i 3 In Exercises 65–68, use the Quadratic Formula to solve the quadratic equation. 65. 67. x2 6x 12 0 3x2 6x 12 0 x2 5x 7 0 66. 68. 8x2 4x 3 0 333202_0503.qxd 12/5/05 9:03 AM Page 389 5.3 Solving Trigonometric Equations Section 5.3 Solving Trigonometric Equations 389 What you should learn • Use standard algebraic techniques to solve trigonometric equations. • Solve trigonometric equations of quadratic type. • Solve trigonometric equations involving multiple angles. • Use inverse trigonometric functions to solve trigonometric equations. Why you should learn it You can use trigonometric equations to solve a variety of real-life problems. For instance, in Exercise 72 on page 398, you can solve a trigonometric equation to help answer questions about monthly sales of skiing equipment. Tom Stillo/Index Stock Imagery Introduction To solve a trigonometric equation, use standard algebraic techniques such as collecting like terms and factoring. Your preliminary goal in solving a trigonometric equation is to isolate the trigonometric function involved in the equation. 2 sin x 1, For example, to solve the equation divide each side by 2 to obtain sin x 1 2. x, x 56 To solve for note in Figure 5.3 that the equation 0, 2. and there are infinitely many other solutions, which can be written as sin x 1 2 sin x Moreover, because in the interval has solutions x 6 2, has a period of x 6 2n and x 5 6 2n General solution where n is an integer, as shown in Figure 5.3 = sin x FIGURE 5.3 Another way to show that the equation sin x 1 2 solutions is indicated in Figure 5.4. Any angles that are coterminal with 56 will also be solutions of the equation. has infinitely many or 6 π 5 sin + 2 6 ( ) π =n 1 2 π 5 6 π sin + 2 6 ( ) π =n 1 2 π 6 FIGURE 5.4 When solving trigonometric |
equations, you should write your answer(s) using exact values rather than decimal approximations. 333202_0503.qxd 12/5/05 9:03 AM Page 390 390 Chapter 5 Analytic Trigonometry Example 1 Collecting Like Terms Solve sin x 2 sin x. Solution Begin by rewriting the equation so that equation. sin x 2 sin x sin x sin x 2 0 sin x sin x 2 2 sin x 2 sin x is isolated on one side of the Write original equation. Add sin x to each side. Subtract 2 from each side. Combine like terms. Divide each side by 2. sin x 2 2 2, and sin x Because x 74. These solutions are each of these solutions to get the general form has a period of x 54 first find all solutions in the interval Finally, add multiples of 0, 2. 2 to x 5 4 2n and x 7 4 2n General solution where n is an integer. Now try Exercise 7. Example 2 Extracting Square Roots Solve 3 tan2 x 1 0. Solution Begin by rewriting the equation so that equation. tan x is isolated on one side of the 3 tan2 x 1 0 3 tan2 x 1 tan2 x 1 3 tan x ± 1 3 ± 3 3, and tan x has a period of x 6 Because These solutions are of these solutions to get the general form x 5 6 n x and 6 where n is an integer. Now try Exercise 11. Write original equation. Add 1 to each side. Divide each side by 3. Extract square roots. first find all solutions in the interval x 56. Finally, add multiples of 0,. to each n General solution 333202_0503.qxd 12/5/05 9:03 AM Page 391 Exploration Using the equation from Example 3, explain what would happen if you divided each side of the equation by Is this a correct method to use when solving equations? cot x. y 1 −1 −2 −3 − π π x y = cot x cos 2 x − 2 cot x FIGURE 5.5 Section 5.3 Solving Trigonometric Equations 391 The equations in Examples 1 and 2 involved only one trigonometric function. When two or more functions occur in the same equation, collect all terms on one side and try to separate the functions by factoring or by using appropriate identities. This may produce factors that yield no solutions, as illustrated in Example 3. Example 3 Fact |
oring Solve cot x cos2 x 2 cot x. Solution Begin by rewriting the equation so that all terms are collected on one side of the equation. cot x cos2 x 2 cot x Write original equation. cot x cos2 x 2 cot x 0 cot xcos2 x 2 0 Subtract 2 cot x from each side. Factor. By setting each of these factors equal to zero, you obtain cot x 0 x 2 and cos2 x 2 0 cos2 x 2 cos x ± 2. cot x 0 The equation solution is obtained for cosine function. Because is obtained by adding multiples of x 2 ± 2 because, has a period of x 2, to has the solution cos x ± 2 cot x to get [in the interval No are outside the range of the the general form of the solution 0, ]. x 2 n General solution is an integer. You can confirm this graphically by sketching the graph of as shown in Figure 5.5. From the graph you can see and so on. These intercepts occur at 32, 2, 2, x- n where y cot x cos 2 x 2 cot x, that the x- 32, cot x cos2 x 2 cot x 0. intercepts correspond to the solutions of Now try Exercise 15. Equations of Quadratic Type Many trigonometric equations are of quadratic type a couple of examples. ax2 bx c 0. Here are Quadratic in sin x 2 sin2 x sin x 1 0 2sin x2 sin x 1 0 Quadratic in sec x sec2 x 3 sec x 2 0 sec x2 3sec x 2 0 To solve equations of this type, factor the quadratic or, if this is not possible, use the Quadratic Formula. 333202_0503.qxd 12/5/05 9:03 AM Page 392 392 Chapter 5 Analytic Trigonometry Example 4 Factoring an Equation of Quadratic Type Find all solutions of 2 sin2 x sin x 1 0 in the interval 0, 2. Algebraic Solution Begin by treating the equation as a quadratic in factoring. sin x and 2 sin2 x sin x 1 0 2 sin x 1sin x 1 0 Write original equation. Factor. Setting each factor equal to zero, you obtain the following solutions in the interval 0, 2. 2 sin x 1 0 and sin x 1 0 sin x |
1 2 x 7, 6 11 6 sin x 1 x 2 Graphical Solution Use a graphing utility set in radian mode to graph y 2 sin2 x sin x 1 as shown in Figure 5.6. Use the zero or root feature or the zoom and trace xfeatures to approximate the 0 ≤ x < 2, intercepts to be for x 1.571, 2 x 3.665 7, 6 and x 5.760 11. 6 These values are 2 sin2 x sin x 1 0 the approximate in the interval 0, 2. solutions of y = 2 sin2x − sin x − 1 2 3 0 −2 Now try Exercise 29. FIGURE 5.6 Example 5 Rewriting with a Single Trigonometric Function Solve 2 sin2 x 3 cos x 3 0. Solution This equation contains both sine and cosine functions. You can rewrite the equa1 cos 2 x. tion so that it has only cosine functions by using the identity sin2 x 2 sin2 x 3 cos x 3 0 21 cos 2 x 3 cos x 3 0 2 cos 2 x 3 cos x 1 0 2 cos x 1cos x 1 0 Write original equation. Pythagorean identity Multiply each side by 1. Factor. Set each factor equal to zero to find the solutions in the interval 5 3 cos x 1 2 2 cos x 1 0, x 3 0, 2. cos x 1 0 x 0 cos x 1 2, the general form of the solution is obtained by cos x Because adding multiples of has a period of 2 to get x 2n, x 2n, 3 where n is an integer. Now try Exercise 31. x 5 3 2n General solution 333202_0503.qxd 12/5/05 9:03 AM Page 393 Section 5.3 Solving Trigonometric Equations 393 Sometimes you must square each side of an equation to obtain a quadratic, as demonstrated in the next example. Because this procedure can introduce extraneous solutions, you should check any solutions in the original equation to see whether they are valid or extraneous. Example 6 Squaring and Converting to Quadratic Type Find all solutions of cos x 1 sin x in the interval 0, 2. Solution It is not clear how to rewrite this equation in terms of a single trigonometric function. Notice what happens when you square each side of the equation. cos x 1 sin x cos 2 x 2 cos x 1 sin2 x cos 2 x 2 |
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