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cos x 1 1 cos 2 x cos 2 x cos2 x 2 cos x 1 1 0 2 cos2 x 2 cos x 0 2 cos xcos x 1 0 Setting each factor equal to zero produces Write original equation. Square each side. Pythagorean identity Rewrite equation. Combine like terms. Factor. 2 cos x 0 cos x 0 x, 2 and cos x 1 0 cos x 1 x. 3 2 Because you squared the origina... |
53 are the only 3t 3 2n and 3t 5 3 2n. Dividing these results by 3, you obtain the general solution t 9 2n 3 and t 5 9 2n 3 General solution where n is an integer. Now try Exercise 35. Example 8 Functions of Multiple Angles Solve 3 tan x 2 3 0. Solution 3 tan x 2 3 0 3 tan tan 3 x 2 x 2 0,, 1 In the interval general, ... |
5/05 9:03 AM Page 396 396 Chapter 5 Analytic Trigonometry 5.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The equation 2 sin 1 0 has the solutions 7 6 2n and 11 6 2n, which are called ________ solutions. 2. The equation 2 tan2 x 3 tan x 1 0 is a trigonometric equation that is of ________ type. 3. A solution to a... |
the multiple-angle equation. 35. 37. 39. cos 2x 1 2 tan 3x 1 x 2 cos 2 2 36. sin 2x 3 2 38. 40. sec 4x 2 x 2 sin 3 2 In Exercises 41– 44, find the -intercepts of the graph. x 41. y sin x 2 1 42. y sin x cos x y 3 2 1 −2 −1 1 2 3 4 −2 43. 3 y tan2x 44. 4 y sec4x 8 y 2 1 −3 −1 1 3 −2 x x x x In Exercises 21–34, find all... |
use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval and (b) solve the trigonometric equation and demonstrate that its x -coordinates of the maximum and solutions are the f. minimum points of (Calculus is required to find the trigonometric equation.) ... |
equilibrium y 0 y 1 12 for y t 74. Projectile Motion A sharpshooter intends to hit a target at a distance of 1000 yards with a gun that has a muzzle velocity of 1200 feet per second (see figure). Neglecting air resistance, determine the gun’s minimum angle of elevation if the range is given by r Equilibrium y r 1 32 2... |
0 θ r = 300 ft Not drawn to scale 76. Data Analysis: Unemployment Rate The table in the United States shows the unemployment rates t for selected years from 1990 through 2004. The time corresponding to is measured in years, with 1990. (Source: U.S. Bureau of Labor Statistics) t 0 r Time, t Rate, r Time, t Rate, r 0 2 4... |
page 405, you can use an identity to rewrite a trigonometric expression in a form that helps you analyze a harmonic motion equation. Using Sum and Difference Formulas In this and the following section, you will study the uses of several trigonometric identities and formulas. Sum and Difference Formulas sinu v sin u co... |
sin 42 cos 12 cos 42 sin 12. Solution Recognizing that this expression fits the formula for sinu v, you can write sin 42 cos 12 cos 42 sin 12 sin42 12 sin 30 1 2. Now try Exercise 31. Example 4 An Application of a Sum Formula Write cosarctan 1 arccos x as an algebraic expression. Solution This expression fits the form... |
x + + sin FIGURE 5.8 You can confirm this graphically by sketching the graph of y sinx sinx 4 1 4 for 0 ≤ x < 2, as shown in Figure 5.8. From the graph you can see that the and 74. x- intercepts are 54 Now try Exercise 69. The next example was taken from calculus. It is used to derive the derivative of the sine functi... |
cos cos (b) sin (b) sin sin sin (b) sin 315 sin 60 3 5 6 3 4 3 4 7 6 In Exercises 7–22, find the exact values of the sine, cosine, and tangent of the angle by using a sum or difference formula. 165 135 30 255 300 45 7 12 4 3 7. 9. 11. 13. 15. 17. 19. 21. 6 5 6 105 60 45 195 225 30 11 3 4 12 9 17 4 12 285 165 13 12 13 ... |
from its equilibrium position, and this motion is modeled by y 1 3 sin 2t 1 4 cos 2t cos x sin x cos x 3 sin x sin x 1 2 6 cos5 x 2 2 4 cos sin 0 2 1 tan tan 1 tan 4 cosx y cosx y cos2 x sin2 y sinx y sinx y) sin2 x sin2 y sinx y sinx y 2 sin x cos y cosx y cosx y 2 cos x cos y 57. 58. 59. 60. 61. 62. 63. 64. In Exerc... |
statement is true or false. Justify your answer. In Exercises 77–80, determine whether the 77. 78. 79. sinu ± v sin u ± sin v cosu ± v cos u ± cos v sin x cosx 2 sinx 80. cos x 2 n 81. 82. is an integer In Exercises 81–84, verify the identity. cosn 1n cos, sinn 1n sin, a sin B b cos Ba2 b2 sinB C, where a sin B b cos ... |
6 g and define the functions and as follows. f h cos6 h cos6 h sin cos h 1 sin h h h gh cos 6 6 (a) What are the domains of the functions and f g? (b) Use a graphing utility to complete the table. 0.01 0.02 0.05 0.1 0.2 0.5 h f h gh (a) Write a proof of the formula for (b) Write a proof of the formula for sinu v. sinu... |
double-angle formulas because they are used often in trigonometry and calculus. For proofs of the formulas, see Proofs in Mathematics on page 425. Double-Angle Formulas sin 2u 2 sin u cos u tan 2u 2 tan u 1 tan2 u cos 2u cos2 u sin2 u 2 cos 2 u 1 1 2 sin2 u Example 1 Solving a Multiple-Angle Equation Solve 2 cos x sin... |
each of the double-angle formulas, you can write sin yr 1213. Consequently, using sin 2 2 sin cos 212 13 cos 2 2 cos2 1 2 25 169 tan 2 sin 2 cos 2 120 119. 120 5 169 13 1 119 169 FIGURE 5.10 Now try Exercise 23. The double-angle formulas are not restricted to angles and. and Other are also valid. Here are two and 3, 6... |
forms of the power-reducing formulas by replacing with The results are called half-angle formulas. u2. u Half-Angle Formulas ±1 cos u ±1 cos u cos sin u 2 2 u 2 2 tan u 2 1 cos u sin u sin u 1 cos u The signs of sin u 2 and cos u 2 depend on the quadrant in which u 2 lies. Example 6 Using a Half-Angle Formula Find the... |
0, 2 to be x- x 0, x 1.571, 2 and x 4.712 3. 2 These values are the approximate solutions of 2 sin2 x 2 cos2x2 0 interval 0, 2. the in 3 2( ) y = 2 − sin2x − 2 cos 2 x − 2 −1 FIGURE 5.11 2 Product-to-Sum Formulas Each of the following product-to-sum formulas is easily verified using the sum and difference formulas dis... |
-Angle and Product-to-Sum Formulas 413 Example 10 Solving a Trigonometric Equation Solve sin 5x sin 3x 0. Solution 2 sin5x 3x 2 sin 5x sin 3x 0 cos5x 3x 0 2 2 sin 4x cos x 0 Write original equation. Sum-to-product formula Simplify. 2 sin 4x equal to zero, you can find that the solutions in the By setting the factor 0, ... |
0 r 1 32 200 1 32 200 200 sin 2 1 sin 2 802 sin 2 Write projectile motion model. Substitute 200 for and 80 for r v0. Simplify. Divide each side by 200. You know that 4 45, Because 45 at an angle of c. From the model 2 2, so dividing this result by 2 produces 4. you can conclude that the player must kick the football so... |
sin4 x cos4 x sin4 x cos2 x In Exercises 35–40, use the figure to find the exact value of the trigonometric function. 1 θ 4 2. 4. 6. 8. tan sin 2 sec 2 cot 2 1. 3. 5. 7. sin cos 2 tan 2 csc 2 In Exercises 9–18, find the exact solutions of the equation in the interval [0, 2. 9. 11. 13. 15. 17. sin 2x sin x 0 4 sin x co... |
2 1 cos 6x 1 cos 8x 1 cos 8x 56. 58. 1 cos 4x 1 cosx 1 2 2 In Exercises 59–62, find all solutions of the equation in [0, 2. the interval Use a graphing utility to graph the equation and verify the solutions. 59. sin x 2 61. cos x 2 cos x 0 sin x 0 60. sin x 2 62. tan x 2 cos x 1 0 sin x 0 In Exercises 63–74, use the p... |
AM Page 417 100. sin cos 3 3 1 sin 2 2 3 101. 102. 103. 1 cos 10y 2 cos 2 5y cos 3 cos u 2 1 4 sin2 ± 2 tan u sec tan u sin u 104. tan u 2 csc u cot u Section 5.5 Multiple-Angle and Product-to-Sum Formulas 417 120. Geometry The length of each of the two equal sides of an isosceles triangle is 10 meters (see figure). T... |
second is 2 sin 2 v0 r where is measured in feet. An athlete throws a javelin at 75 feet per second. At what angle must the athlete throw the javelin so that the javelin travels 130 feet? of 1. (b) Find the angle that corresponds to a mach number of 4.5. (c) The speed of sound is about 760 miles per hour. Determine th... |
graphing utility to rule out incorrectly rewritten functions. (d) Rewrite the result of part (c) in terms of the sine of a double angle. Use a graphing utility to rule out incorrectly rewritten functions. (e) When you rewrite a trigonometric expression, the result may not be the same as a friend’s. Does this mean that... |
382). Section 5.3 Use standard algebraic techniques to solve trigonometric equations (p. 389). Solve trigonometric equations of quadratic type (p. 391). Solve trigonometric equations involving multiple angles (p. 394). Use inverse trigonometric functions to solve trigonometric equations (p. 395). Section 5.4 Use sum a... |
2 1 csc 1 1 csc 1 22. cos2 x 1 sin x 23. Rate of Change The rate of change of the function the expression Show that this expression can also be is given by f x csc x cot x csc2 x csc x cot x. written as 1 cos x sin2 x. 5.2 In Exercises 25–32, verify the identity. 25. 26. 27. 28. 29. 30. 31. 32. cos xtan2 x 1 sec x sec2... |
05 9:07 AM Page 421 In Exercises 55–58, write the expression as the sine, cosine, or tangent of an angle. 55. 56. 57. sin 60 cos 45 cos 60 sin 45 cos 45 cos 120 sin 45 sin 120 tan 25 tan 10 1 tan 25 tan 10 58. tan 68 tan 115 1 tan 68 tan 115 In Exercises 59–64, find the exact value of the trigonometsin u 3 u ric functi... |
the half-angle formulas. tanu/2 5, 0 < u < 2 8, < u < 32 and sin u 3 tan u 5 cos u 2 7, 2 < u < sec u 6, 2 < u < 88. 89. 90. In Exercises 91 and 92, use the half-angle formulas to simplify the expression. 91. 1 cos 10x 2 92. sin 6x 1 cos 6x In Exercises 93–96, use the product-to-sum formulas to write the product as a ... |
of the weight. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 107–110, determine whether 107. If < <, then cos < 0. 2 2 sinx y sin x sin y 4 sinx cosx 2 sin 2x 4 sin 45 cos 15 1 3 108. 109. 110. 111. List the reciprocal identities, quotient identities, and Pythagorean identi... |
the values of, 6. Use a graphing utility to graph the functions Make a conjecture about y1 0 ≤ < 2, for which y1 and Verify the result algebraically. tan sec2 1 y2 and cos x sin x tan x y2. is true. sec x. In Exercises 7–12, verify the identity. 7. 9. 11. 12. sin sec tan csc sec sin cos sinn 1n sin, n sin x cos x2 1 s... |
Proofs in Mathematics Sum and Difference Formulas (p. 400) sinu v sin u cos v cos u sin v sinu v sin u cos v cos u sin v cosu v cos u cos v sin u sin v cosu v cos u cos v sin u sin v tanu v tan u tan v 1 tan u tan v tanu v tan u tan v 1 tan u tan 1, 0 ) Proof You can use the figures at the left for the proofs of the f... |
identity Sum and difference formulas Divide numerator and denominator by cos u cos v. D x y = (, 3 3 ) 424 333202_050R.qxd 12/5/05 9:07 AM Page 425 sin u cos v cos u cos v cos u cos v cos u cos v ± cos u sin v cos u cos v sin u sin v cos u cos v Write as separate fractions. sin u cos u ± sin v cos v sin v cos v 1 sin ... |
1 cos 2u Sum-to-Product Formulas sin u sin v 2 sinu v 2 sin u sin v 2 cosu v 2 cos u cos v 2 cosu v 2 cos u cos v 2 sinu v 2 (p. 412) cosu v 2 sinu v 2 cosu v 2 sinu v 2 Proof To prove the first formula, u x y2 and x u v in the product-to-sum formula. y u v. and Then substitute let v x y2 sin u cos v 1 2 1 2 sinu v si... |
for the path sin, cos, and tan 2 2 2 t 0.006 where is an acute angle. −1.4 (b) Find the period of each sine component of p. Is p periodic? If so, what is its period? (c) Use the zero or root feature or the zoom and trace features of a graphing utility to find the -intercepts of the graph of over one cycle. p t (d) Use... |
prism made of glass. 13. (a) Write a sum formula for (b) Write a sum formula for sinu v w. tanu v w. 14. (a) Derive a formula for (b) Derive a formula for cos 3. cos 4. h 15. The heights engine can be modeled by (in inches) of pistons 1 and 2 in an automobile h1 3.75 sin 733t 7.5 and h2 3.75 sin 733t 4 3 7.5 where t i... |
Vectors in the Plane Vectors and Dot Products Trigonometric Form of a Complex Number 66 The work done by a force, such as pushing and pulling objects, can be calculated using vector operations AT I O N S Triangles and vectors have many real-life applications. The applications listed below represent a small sample of t... |
cases require the Law of Cosines (see Section 6.2). Law of Sines If ABC is a triangle with sides a, b, and c, then Hideo Kurihara/Getty Images a sin A b sin B c sin is acute. A is obtuse. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter... |
43 23.84 feet. sin 39 Now try Exercise 35. For practice, try reworking Example 2 for a pole that tilts away from the sun under the same conditions. 333202_0601.qxd 12/5/05 10:40 AM Page 432 432 Chapter 6 Additional Topics in Trigonometry The Ambiguous Case (SSA) In Examples 1 and 2 you saw that two angles and one side... |
B bsin A a sin B 25sin 85 15 1.660 > 1 This contradicts the fact that and sides a 15 b 25 sin B ≤ 1. and an angle of Reciprocal form Multiply each side by b. So, no triangle can be formed having A 85. Now try Exercise 21. Example 5 Two-Solution Case—SSA Find two triangles for which a 12 meters, b 31 meters, and A 20.5... |
explaining how to use the Law of Sines to solve each triangle. Is there an easier way to solve these triangles? a. AAS B 50° c = 20 b. ASA B 50° a = 10 C A C A 333202_0601.qxd 12/5/05 10:40 AM Page 436 436 Chapter 6 Additional Topics in Trigonometry 6.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text co... |
36, A 60, A 10, A 88, a 5 a 10 a 10.8 a 315.6 In Exercises 29–34, find the area of the triangle having the indicated angle and sides. a 8, a 9, A 36, A 60, A 102.4, A 24.3, A 83 20, A 5 40, B 15 30, B 2 45, C 145, C 16.7, C 54.6, a 21.6 c 2.68 C 54.6, c 18.1 b 4.8 B 8 15, a 4.5, b 6.2, b 4, c 14 b 6.8 c 5.8 29. 30. 31... |
known quantities on the triangle and use a variable to indicate the height of the flagpole. (b) Write an equation involving the unknown quantity. (c) Find the height of the flagpole. 37. Angle of Elevation A 10-meter telephone pole casts a 17-meter shadow directly down a slope when the angle of the angle of elevation ... |
boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to the lighthouse is S E, and 15 minutes later the bearing is S E (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline? 63 70 70° d 63° W N S E Model... |
region and the domain of the function would change if the eight-centimeter line segment were decreased in length. 10 20 30 40 50 60 Skills Review d In Exercises 49–52, use the fundamental trigonometric identities to simplify the expression. 49. 51. sin x cot x 1 sin2 2 x 50. 52. 1 cot2 2 tan x cos x sec x x 333202_060... |
the Law of Sines to determine sin A asin B b is an obtuse angle given by A. 8sin 116.80 0.37583 19 B B Because obtuse angle. So, A A 22.08 is obtuse, must be acute, because a triangle can have, at most, one and C 180 22.08 116.80 41.12. Now try Exercise 1. 333202_0602.qxd 12/5/05 10:41 AM Page 440 440 Chapter 6 Additi... |
12/5/05 10:41 AM Page 441 Section 6.2 Law of Cosines 441 Applications Example 3 An Application of the Law of Cosines The pitcher’s mound on a women’s softball field is 43 feet from home plate and the distance between the bases is 60 feet, as shown in Figure 6.13. (The pitcher’s mound is not halfway between home plate ... |
a triangle. This formula is called Heron’s Area Formula after the Greek mathematician Heron (c. 100 B.C.). Heron’s Area Formula Given any triangle with sides of lengths triangle is Area ss as bs c where s a b c2. a, b, and c, the area of the For a proof of Heron’s Area Formula, see Proofs in Mathematics on page 491. E... |
.5 A a = 10 c B C b = 10 A c = 15 a = 7 B b = 15 30° A C a c = 30 B c 20 c 72 b 14, b 25, a 11, a 55, a 75.4, a 1.42, A 135, A 55, B 10 35, B 75 20, B 125 40, C 15 15, C 43, C 103, c 52 c 1.25 b 52, b 0.75, b 4, b 3, c 9 c 10 c 30 c 9.5 c 32 b 2.15 a 40, a 6.2, a 32, a 6.25, b 7 a 4 9, 9 a 3 b 3 8, 4 In Exercises 17–22... |
. Surveying To approximate the length of a marsh, a B, then (see figure). surveyor walks 250 meters from point turns Approximate the length and walks 220 meters to point AC of the marsh. to point C 75 A B 75° 220 m 250 m C A 32. Surveying A triangular parcel of land has 115 meters of frontage, and the other boundaries ... |
the wall 420 feet from the camera (see figure). The camera turns to follow the play. Approximately how far does the center fielder have to run to make the catch? 8 100 ft 6° 75 f t 75 f t 330 ft 8° 420 ft 333202_0602.qxd 12/8/05 9:22 AM Page 445 Section 6.2 Law of Cosines 445 41. Aircraft Tracking To determine the dis... |
are approximately 200 feet, 500 feet, and 600 feet. Approximate the area of the parcel. 48. Geometry A parking lot has the shape of a parallelogram (see figure). The lengths of two adjacent sides are 70 meters and 100 meters. The angle between the two sides is 70 What is the area of the parking lot?. (c) Use a graphin... |
ises 55 and 55. Given a triangle with a 25, b 55, and c 72 find the areas of (a) the triangle, (b) the circumscribed circle, and (c) the inscribed circle. 56. Find the length of the largest circular running track that can be built on a triangular piece of property with sides of lengths 200 feet, 250 feet, and 325 feet.... |
Formula. has initial point \ PQ Q. P \ Terminal point Q PQ P Initial point FIGURE 6.15 FIGURE 6.16 Two directed line segments that have the same magnitude and direction are equivalent. For example, the directed line segments in Figure 6.16 are all equivalent. The set of all directed line segments that are equivalent t... |
2 is given by v1, v2 p2 v The magnitude (or length) of is given by 2 q2 p2 is a unit vector. Moreover, v q1 v 1, p1 v v. 2 v1 2 v2 2. If vector 0. v 0 if and only if v is the zero u2 and v v1, v2 u u1, u2 Two vectors v2. u PQ For instance, in Example 1, the vector \ 3 0, 2 0 3, 2 R 1, 2 S 4, 4 and the vector v RS \ v f... |
k To add two vectors geometrically, position them (without changing their lengths or directions) so that the initial point of one coincides with the terminal is formed by joining the initial point of the secpoint of the other. The sum ond vector with the terminal point of the first vector, as shown in Figure 6.20. Thi... |
Figure 6.23. Note that the figure shows the w v as the sum w v. c. The sum of and 2 is w v 2w 2, 5 23, 4 v 2, 5 23, 24 2, 5 6, 8 2 6, 5 8 4, 13. v 2w A sketch of is shown in Figure 6.24. y y (3, 4) w −v 4 3 2 1 − ( 4, 10) − ( 2, 5) 10 8 2v 6 4 v −8 −6 −4 −2 2 x −1 FIGURE 6.22 FIGURE 6.23 x w − v 3 4 5 (5, −1) Now try ... |
29 25 29 29 29 1. Now try Exercise 31 Historical Note William Rowan Hamilton (1805–1865), an Irish mathematician, did some of the earliest work with vectors. Hamilton spent many years developing a system of vector-like quantities called quaternions. Although Hamilton was convinced of the benefits of quaternions, the o... |
solve this problem by converting to component form. This, however, is not necessary. It is just as easy to perform the operations in unit vector form. and u v 2u 3v 23i 8j 32i j 6i 16j 6i 3j 12i 19j Now try Exercise 49. 333202_0603.qxd 12/8/05 9:26 AM Page 453 y 1 x y (, ) y = sin θ u θ −1 x = cos θ 1 x −1 FIGURE 6.27... |
AM Page 454 454 Chapter 6 Additional Topics in Trigonometry Applications of Vectors − 100 −75 −50 210° −50 −75 100 FIGURE 6.30 W 15° B C D 15° A FIGURE 6.31 y Example 8 Finding the Component Form of a Vector Find the component form of the vector that represents the velocity of an airplane descending at a speed of 100 ... |
b) xx Solution Using Figure 6.32, the velocity of the airplane (alone) is v1 500cos 120, sin 120 250, 2503 and the velocity of the wind is 70cos 45, sin 45 352, 352. v2 So, the velocity of the airplane (in the wind) is v2 v v1 250 352, 2503 352 200.5, 482.5 and the resultant speed of the airplane is v 200.52 482.52 mil... |
x y 4 (0, 4) u −2 −4 v (3, 3) x 2 4 −4 (−3, −4) (0, −5) In Exercises 3–14, find the component form and the magnitude of the vector v. 3. y 4 3 2 1 5. (3, 2) x 3 4 v 1 2 y (−1, 4) 5 v 3 2 1 (2, 2) −3 −2 −1 21 3 x 4. 6. y 1 −1 −2 −3 x −4 −3 −2 v − − ( 4, 2) (3, 5) y 6 4 2 v −4 −2 (−1, −1) x 2 4 7. y 8. y 4 3 2 1 (3, 3) ... |
29–38, find a unit vector in the direction of the given vector. 29. 31. 33. 35. 37. u 3, 0 v 2, 2 v 6i 2j w 4j w i 2j 30. 32. 34. 36. 38. u 0, 2 v 5, 12 v i j w 6i w 7j 3i In Exercises 39– 42, find the vector v with the given magnitude and the same direction as u. Magnitude v 5 v 6 v 9 v 10 39. 40. 41. 42. Direction u... |
ant Force In Exercises 71 and 72, find the angle between the forces given the magnitude of their resultant. (Hint: Write force 1 as a vector in the direction of the positive -axis and force 2 as a vector at an angle with the positive -axis.) x x Force 1 71. 45 pounds Force 2 60 pounds Resultant Force 90 pounds 72. 3000... |
). Find the tension in the tow lines if they each make an angle with the axis of the barge. 18 18° 18° 82. Rope Tension To carry a 100-pound cylindrical weight, two people lift on the ends of short ropes that are tied to an eyelet on the top center of the cylinder. Each rope makes a 20 angle with the vertical. Draw a f... |
same magnitude and direction, then u u v. Section 6.3 Vectors in the Plane 459 (b) If the resultant of the forces is make a conjecture 0, about the angle between the forces. (c) Can the magnitude of the resultant be greater than the sum of the magnitudes of the two forces? Explain. 90. Graphical Reasoning Consider two... |
_0604.qxd 12/5/05 10:44 AM Page 460 460 Chapter 6 Additional Topics in Trigonometry 6.4 Vectors and Dot Products What you should learn • Find the dot product of two vectors and use the Properties of the Dot Product. • Find the angle between two vectors and determine whether two vectors are orthogonal. • Write a vector ... |
product. Solution Begin by finding the dot product of and u v. u v 1, 3 2, 4 12 34 14 u vw 141, 2 14, 28 u 2v 2u v 214 28 a. b. Notice that the product in part (a) is a vector, whereas the product in part (b) is a scalar. Can you see why? Now try Exercise 11. Example 3 Dot Product and Magnitude u The dot product of wi... |
< 2 0 < cos < 1 Acute Angle v u 0 cos 1 Same Direction Definition of Orthogonal Vectors The vectors and are orthogonal if u v 0. u v The terms orthogonal and perpendicular mean essentially the same thing—meeting at right angles. Even though the angle between the zero vector and another vector is not defined, it is con... |
464 464 Chapter 6 Additional Topics in Trigonometry Definition of Vector Components Let and be nonzero vectors such that u v u w1 w1 w2 w2 w1 u projvu. w2 and are orthogonal and w1 where v, as shown in Figure 6.38. The vectors nents of is parallel to (or a scalar multiple of) w1 are called vector compo- w2 and is the ... |
. Now try Exercise 53. Example 7 Finding a Force A 200-pound cart sits on a ramp inclined at 30, as shown in Figure 6.40. What force is required to keep the cart from rolling down the ramp? Solution Because the force due to gravity is vertical and downward, you can represent the gravitational force by the vector F 200j... |
work as follows. Projection form for work W proj PQ \ \F PQ cos 60F PQ \ 1 2 5012 300 foot-pounds So, the work done is 300 foot-pounds. You can verify this result by finding the vectors and calculating their dot product. and PQ F \ 12 ft P projPQF 60° Q F 12 ft FIGURE 6.43 Now try Exercise 69. 333202_0604.qxd 12/5/05 ... |
39–42, use vectors to find the interior angles of the triangle with the given vertices. 39. 41. 1, 2, 3, 4, 2, 5 3, 0, 2, 2, 0, 6) 40. 42. 3, 4, 3, 5, 1, 7, 1, 9, 8, 2 7, 9 where is the angle between In Exercises 9–18, use the vectors and whether the result is a vector or a scalar. u <2, 2>, v <3, 4>, to find the indi... |
is u projv u. v u 53. 55. u 2, 2 v 6, 1 u 0, 3 v 2, 15 54. 56. u 4, 2 v 1, 2 u 3, 2 v 4, 1 In Exercises 57 and 58, use the graph to determine mentally the projection of onto. (The coordinates of the terminal points of the vectors in standard position are given.) Use the formula for the projection of onto to verify you... |
keep the truck from rolling down the hill in terms of the slope d. (b) Use a graphing utility to complete the table 10 d Force d Force (c) Find the force perpendicular to the hill when d 5. 68. Braking Load A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of Assume that the only force to... |
What is known about v between two nonzero vectors condition? and u, the angle, under each (a) u v 0 u v > 0 78. Think About It What can be said about the vectors and u v < 0 (b) (c) u v under each condition? (a) The projection of onto equals u v u. (b) The projection of onto equals u v 0. 79. Proof Use vectors to prov... |
.44. Imaginary axis 3 2 1 (3, 1) or 3 + i 1 2 3 Real axis −3 −2 −1 (−2, −1) or −2 − i −1 −2 FIGURE 6.44 The absolute value of the complex number between the origin 0, 0 and the point a, b. a bi is defined as the distance Definition of the Absolute Value of a Complex Number The absolute value of the complex number z a b... |
there are infinitely many choices for complex number is not unique. Normally, 0 ≤ < 2, although on occasion it is convenient to use < 0., Example 2 Writing a Complex Number in Trigonometric Form Write the complex number z 2 23i in trigonometric form. Solution The absolute value of z is r 2 23i 22 232 16 4 Imaginary ax... |
cos 2 r2 r1 z2 be complex z1z2 z1 z2 r1r2 r1 r2 cos 1 2 i sin 1 2 cos 1 2 i sin 1 2, z 2 0 Product Quotient Note that this rule says that to multiply two complex numbers you multiply moduli and add arguments, whereas to divide two complex numbers you divide moduli and subtract arguments. 333202_0605.qxd 12/5/05 10:45 ... |
2 z3 r 2cos 2 i sin 2rcos i sin r 3cos 3 i sin 3 z4 r 4cos 4 i sin 4 z5 r5cos 5 i sin 5... This pattern leads to DeMoivre’s Theorem, which is named after the French mathematician Abraham DeMoivre (1667–1754). DeMoivre’s Theorem z rcos i sin If then is a complex number and n is a positive integer, zn rcos i sin n rncos... |
complex number are useful for solving some polynomial equations. For instance, explain how you can use DeMoivre’s Theorem to solve the polynomial equation x4 16 0. 16 [Hint: Write as 16cos i sin. ] To find a formula for an th root of a complex number, let be an th root n n u of where z, u scos i sin and z rcos i sin. ... |
a Real Number Find all the sixth roots of 1. Solution First write 1 in the trigonometric form root formula, with 61cos r 1, 0 2k 6 0 2k 6 i sin n 6 and 1 1cos 0 i sin 0. the roots have the form k 3 cos i sin k 3 Then, by the th n. So, for 1, 2, 3, 4, and 5, the sixth roots are as follows. (See Figure 6.49.) k 0, cos 0... |
103. W RITING ABOUT MATHEMATICS A Famous Mathematical Formula The famous formula Imaginary axis −1.3660 + 0.3660i 1 −1 −2 −2 FIGURE 6.50 is Note in Example 8 that the z absolute value of r 2 2i 22 22 8 and the angle tan b a 2 2 1. is given by ea bi e acos b i sin b is called Euler’s Formula, after the Swiss mathematic... |
Real axis 3 = 3 − i z −3 10. Imaginary axis 3 i = 1 + 3 − z 25. 27. 29. 3 i 5 2i 8 53 i 26. 28. 30. 1 3i 8 3i 9 210 i In Exercises 31– 40, represent the complex number graphically, and find the standard form of the number. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 3cos 120 i sin 120 5cos 135 i sin 135 cos 300 i sin 300 ... |
45cos 310i sin 310 0.60cos 200 i sin 200 cos 5 i sin 5cos 20 i sin 20 cos 50 i sin 50 cos 20 i sin 20 2cos 120 i sin 120 4cos 40 i sin 40 cos53 i sin53 cos i sin 5cos 4.3 i sin 4.3 4cos 2.1 i sin 2.1 12cos 52 i sin 52 3cos 110 i sin 110 6cos 40 i sin 40 7cos 100 i sin 100 In Exercises 59–66, (a) write the trigonometric... |
of 93. Square roots of 94. Fourth roots of 95. Cube roots of 96. Cube roots of i sin 32cos 25i 625i 1 3 i 125 2 421 i 97. Fourth roots of 16 98. Fourth roots of i 99. Fifth roots of 1 100. Cube roots of 1000 125 4 102. Fourth roots of 101. Cube roots of 103. Fifth roots of 1281 i 104. Sixth roots of 64i 333202_0605.qx... |
1 1 1 show that r2 r1 r2 z1 z 2 cos 1 2 i sin 1 2. B a C 125. 127. 129. and a 8 b 112.6 A 22, A 30, A 42 15, c 11.2 c b 126. 128. 130. A a 33.5 b 211.2 B 66, B 6, B 81 30, c 6.8 Harmonic Motion In Exercises 131–134, for the simple harmonic motion described by the trigonometric function, find the maximum displacement an... |
). Use vectors to model and solve real-life problems (p. 454). Section 6.4 Find the dot product of two vectors and use the properties of the dot product (p. 460). Find the angle between two vectors and determine whether two vectors are orthogonal (p. 461). Write vectors as sums of two vector components (p. 463). Use ve... |
123, A 11, b 5, a 4, a 16, b 22, b 5 c 21 17. Height From a certain distance, the angle of elevation to 17. At a point 50 meters closer to Approximate the the top of a building is the building, the angle of elevation is height of the building. 31. 18. Geometry Find the length of the side w of the parallelogram. w 140°... |
Navigation Two planes leave Raleigh-Durham Airport at approximately the same time. One is flying 425 miles per, and the other is flying 530 miles hour at a bearing of 67. per hour at a bearing of Draw a figure that gives a visual representation of the problem and determine the distance between the planes after they ha... |
. 56. w 2u v w 4u 5v w 3v w 1 2v In Exercises 57– 60, write vector as a linear combination of the standard unit vectors and u j. i 57. 58. 59. u 3, 4 u 6, 8 u has initial point 60. u has initial point 3, 4 2, 7 and terminal point and terminal point 9, 8. 5, 9. In Exercises 61 and 62, write the vector vcos i sin j. v in... |
, find the angle between the vectors. u cos i sin v cos i sin cos 45i sin 45j v cos 300i sin 300j u 22, 4, u 3, 3, v 4, 33 v 2, 1 81. 82. 83. 84. In Exercises 85–88, determine whether orthogonal, parallel, or neither. u and v are 85. 87. u 3, 8 v 8, 3 u i v i 2j 86. 88. u 1 4, 1 2 v 2, 4 u 2i j v 3i 6j In Exercises 89–... |
ises 111–114, (a) use the theorem on page 476 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. v 8, 2 111. Sixth roots of 89. 90. 91. 92. u 4, 3, u 5, 6, u 2, 7, u 3, 5, v 10, 0 v 1, 1 v 5, 2 729i 256i 112. Fourth roots of ... |
° 4 Real axis Real axis C B A x D E 133. The figure shows z1 and Describe z2. z1z2 and z1 z2. Imaginary axis z2 1 θ z1 θ −1 1 Real axis u 128. The vectors and have the same magnitudes in the two figures. In which figure will the magnitude of the sum be greater? Give a reason for your answer. v (a) y (b) y v u v x u x 1... |
. u v 13. 5u 3v 14. Find a unit vector in the direction of u 4, 3. 15. Forces with magnitudes of 250 pounds and 130 pounds act on an object at angles of, respectively, with the -axis. Find the direction and magnitude of the x and 45 60 resultant of these forces. 16. Find the angle between the vectors u 1, 5 and v 3, 2.... |
. hx secx 7. Find a, b, graph in the figure. c and such that the graph of the function hx a cosbx c matches the 8. Sketch the graph of the function f x 1 2x sin x over the interval 3 ≤ x ≤ 3. In Exercises 9 and 10, find the exact value of the expression without using a calculator. 9. tanarctan 6.7 10. tanarcsin 3 5 11.... |
vector u 3, 5 as a linear combination of the standard unit vectors i and j. 32. Find a unit vector in the direction of v i j. u 3i 4j for 33. Find u v 34. Find the projection of orthogonal vectors. and u 8, 2 v i 2j. onto v 1, 5. Then write u as the sum of two 35. Write the complex number 36. Find the product of 2 2i ... |
of Tangents, which was developed by Francois Vi`ete (1540–1603). The Law of Tangents follows from the Law of Sines and the sum-to-product formulas for sine and is defined as follows. a b a b tanA B2 tanA B2 The Law of Tangents can be used to solve a triangle when two sides and the included angle are given (SAS). Befor... |
b sin A. it follows that Furthermore, Because c, 0. and C B a x b cos A where B, to vertex a x c2 y 02 a2 x c2 y 02 a2 b cos A c2 b sin A2 a2 b2 cos2 A 2bc cos A c2 b2 sin2 A a2 b2sin2 A cos2 A c2 2bc cos A a2 b2 c2 2bc cos A. To prove the second formula, consider the bottom triangle at the left, which also A C has th... |
bc1 cos A s bs c. By substituting into the last formula for area, you can conclude that Area ss as bs c. 491 333202_060R.qxd 12/8/05 9:34 AM Page 492 be vectors in the plane or in space and let be a scalar. c 1. Properties of the Dot Product Let v,u, w and. 5. cu v cu v u cv (p. 460) 2. 4. 0 v 0 v v v2 0 0, 0, and let... |
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