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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 calcula... |
.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 ... |
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 trig... |
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 mathem... |
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. Sin... |
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... |
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 ... |
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 si... |
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... |
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... |
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 visua... |
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 ... |
(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 ... |
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... |
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... |
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... |
. 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 ... |
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 ... |
> 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 ... |
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 eq... |
. 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 w... |
. 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 amp... |
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 ... |
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—... |
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... |
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,... |
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 ... |
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 m... |
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 ________ __... |
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... |
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 ... |
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 Tun... |
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 t... |
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 th... |
. 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 ... |
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... |
, 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 a... |
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, ... |
� 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... |
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 an... |
: 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 arithmeticall... |
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. ... |
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 fun... |
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. ... |
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. ... |
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 th... |
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 tr... |
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 Inve... |
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, ... |
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... |
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 arcs... |
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 ... |
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... |
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. ... |
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 ... |
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 alon... |
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... |
? (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 m... |
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 i... |
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.... |
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 w... |
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 positi... |
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 ... |
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 tr... |
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... |
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... |
: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 ... |
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 utilit... |
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 ... |
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 ... |
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 ... |
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 ... |
. 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... |
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. ... |
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 Chapte... |
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... |
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... |
. 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 ... |
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 ... |
(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 t... |
). 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 ... |
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 c... |
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 functi... |
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... |
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 ... |
: 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... |
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... |
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 ... |
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 th... |
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 ... |
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... |
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 crea... |
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 identiti... |
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 ... |
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 ... |
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. sin... |
. 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 Pa... |
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 si... |
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... |
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 ... |
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