Trigonometry 10th edition pdf download

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Chapter 2 Acute Angles and Right Triangles Section 2.1 Trigonometric Functions of Acute Angles 1.

side opposite = hypotenuse side adjacent cos A = = hypotenuse side opposite tan A = = side adjacent sin A =

21 29 20 29 21 20

side opposite 45 2. sin A = = hypotenuse 53 side adjacent 28 cos A = = hypotenuse 53 side opposite 45 tan A = = side adjacent 28 3.

side opposite n = hypotenuse p side adjacent m cos A = = hypotenuse p side opposite n tan A = = side adjacent m sin A =

side opposite k = hypotenuse z side adjacent y cos A = = hypotenuse z side opposite k tan A = = side adjacent y

4. sin A =

For Exercises 5−10, refer to the Function Values of Special Angles chart on page 54 of the text. 5. C; sin 30° =

1 2

6. H; cos 45° =

2

9. E; csc 60° =

1 1 2 = = 3 sin 60° 3 2

2 3 2 3 = ⋅ = 3 3 3 10. A; cot 30° =

cos 30° = sin 30°

3 2 1 2

=

3 2 ⋅ = 3 2 1

11. a = 5, b = 12 c 2 = a 2 + b 2 ⇒ c 2 = 52 + 122 ⇒ c 2 = 169 ⇒ c = 13 side opposite b 12 sin B = = = hypotenuse c 13 side adjacent a 5 cos B = = = hypotenuse c 13 side opposite b 12 tan B = = = side adjacent a 5 side adjacent a 5 cot B = = = side opposite b 12 hypotenuse c 13 sec B = = = side adjacent a 5 hypotenuse c 13 csc B = = = side opposite b 12 12. a = 3, b = 4 c 2 = a 2 + b 2 ⇒ c 2 = 32 + 4 2 ⇒ c 2 = 25 ⇒ c=5 side opposite b 4 sin B = = = hypotenuse c 5 side adjacent a 3 cos B = = = hypotenuse c 5 side opposite b 4 tan B = = = side adjacent a 3 side adjacent a 3 = = cot B = side opposite b 4 hypotenuse c 5 2 sec B = = =

7. B; tan 45° = 1 1

1

side adjacent a 3 hypotenuse c 5 csc B = = = side opposite b 4

8. G; sec 60° =

48

cos 60°

=

=2

1 2

Chapter 2 Acute Angles and Right Triangles

13. a = 6, c = 7 c 2 = a 2 + b 2 ⇒ 7 2 = 62 + b 2 ⇒ 49 = 36 + b 2 ⇒ 13 = b 2 ⇒ 13 = b side opposite b sin B = = = hypotenuse c side adjacent a cos B = = = hypotenuse c side opposite b tan B = = = side adjacent a cot B = = sec B = csc B = =

13 7 6 7 13 6

side adjacent a 6 = = side opposite b 13 6 13 6 13 ⋅ = 13 13 13 hypotenuse c 7 = = side adjacent a 6 hypotenuse c 7 = = side opposite b 13 7 13 7 13 ⋅ = 13 13 13

14. b = 7, c = 12 c 2 = a 2 + b 2 ⇒ 12 2 = a 2 + 7 2 ⇒ 144 = a 2 + 49 ⇒ 95 = a 2 ⇒ 95 = a side opposite b 7 sin B = = = hypotenuse c 12 side adjacent a 95 cos B = = = hypotenuse c 12 side opposite b 7 tan B = = = side adjacent a 95 7 95 7 95 = ⋅ = 95 95 95 side adjacent a 95 = = side opposite b 7 hypotenuse c 12 sec B = = = side adjacent a 95 12 95 12 95 = ⋅ = 95 95 95 hypotenuse c 12 csc B = = = side opposite b 7 cot B =

15. a = 3, c = 10 c 2 = a 2 + b 2 ⇒ 10 2 = 32 + b 2 ⇒ 100 = 9 + b 2 ⇒ 91 = b 2 ⇒ 91 = b side opposite b 91 sin B = = = hypotenuse c 10 side adjacent a 3 cos B = = = hypotenuse c 10

tan B = cot B = = sec B = csc B = =

side opposite b 91 = = side adjacent a 3 side adjacent a 3 = = side opposite b 91 3 91 3 91 ⋅ = 91 91 91 hypotenuse c 10 = = side adjacent a 3 hypotenuse c 10 = = side opposite b 91 10 91 10 91 ⋅ = 91 91 91

16. b = 8, c = 11 c 2 = a2 + b 2 ⇒ 112 = a 2 + 82 ⇒ 121 = a2 + 64 ⇒ 57 = a 2 ⇒ 57 = a side opposite b 8 sin B = = = hypotenuse c 11 side adjacent a 57 cos B = = = hypotenuse c 11 side opposite b 8 tan B = = = side adjacent a 57 8 57 8 57 = ⋅ = 57 57 57 side adjacent a 57 = = side opposite b 8 hypotenuse c 11 sec B = = = side adjacent a 57 11 57 11 57 = ⋅ = 57 57 57 hypotenuse c 11 csc B = = = side opposite b 8 cot B =

17. a = 1, c = 2 c 2 = a 2 + b 2 ⇒ 2 2 = 12 + b 2 ⇒ 4 = 1 + b2 ⇒ 3 = b2 ⇒ 3 = b sin B = cos B = tan B = cot B = =

side opposite = hypotenuse side adjacent = hypotenuse side opposite = side adjacent side adjacent = side opposite 1 3 3 ⋅ = 3 3 3

b c a c b a a b

3 2 1 = 2 3 = = 3 1 1 = 3

=

(continued on next page)

Section 2.1 Trigonometric Functions of Acute Angles

(continued) hypotenuse = side adjacent hypotenuse csc B = = side opposite 2 3 2 3 = ⋅ = 3 3 3 sec B =

c 2 = =2 a 1 c 2 = b 3

2

c 2 = a 2 + b 2 ⇒ 22 = 2 + b 2 ⇒ 4 = 2 + b2 ⇒ 2 = b2 ⇒ 2 = b

cos B = tan B = cot B = sec B = = csc B = =

49

21. cos 30° = sin (90° − 30°) = sin 60° 22. sin 45° = cos (90° − 45°) = cos 45° 23. csc 60° = sec (90° − 60°) = sec 30°

18. a = 2, c = 2

sin B =

20. sin θ = cos (90° − θ ) ; cos θ = sin (90° − θ ) ; tan θ = cot (90° − θ ) ; cot θ = tan (90° − θ ) ; sec θ = csc (90° − θ ) ; csc θ = sec (90° − θ )

side opposite b 2 = = hypotenuse c 2 side adjacent a 2 = = hypotenuse c 2 side opposite b 2 = = =1 side adjacent a 2 side adjacent a 2 = = =1 side opposite b 2 hypotenuse c 2 = = side adjacent a 2 2 2 2 2 ⋅ = = 2 2 2 2 hypotenuse c 2 = = side opposite b 2 2 2 2 2 ⋅ = = 2 2 2 2

19. b = 2, c = 5 c 2 = a 2 + b 2 ⇒ 52 = a 2 + 22 ⇒ 25 = a 2 + 4 ⇒ 21 = a2 ⇒ 21 = a side opposite b 2 sin B = = = hypotenuse c 5 side adjacent a 21 cos B = = = hypotenuse c 5 side opposite b 2 tan B = = = side adjacent a 21 2 21 2 21 = ⋅ = 21 21 21 side adjacent a 21 cot B = = = side opposite b 2 hypotenuse c 5 sec B = = = side adjacent a 21 5 21 5 21 = ⋅ = 21 21 21 hypotenuse c 5 csc B = = = side opposite b 2

24. cot 73° = tan (90° − 73°) = tan17° 25. sec 39° = csc (90° − 39°) = csc 51° 26. tan 25.4° = cot (90° − 25.4°) = cot 64.6° 27. sin 38.7° = cos (90° − 38.7°) = cos 51.3° 28. cos (θ + 20°) = sin ⎡⎣ 90° − (θ + 20°) = sin (70° − θ ) 29. sec (θ + 15° ) = csc ⎡90 ⎣ ° − (θ + 15° )⎤⎦ = csc (75° − θ ) 30. Using θ = 50°, 102°, 248°, and − 26°, we see that sin (90° − θ ) and cos θ yield the same values.

For exercises 31−40, if the functions in the equations are cofunctions, then the equations are true if the sum of the angles is 90º. 31.

32.

tan α = cot (α + 10°) α + (α + 10°) = 90° 2α + 10° = 90° 2α = 80° ⇒ α = 40° cos θ = sin ( 2θ − 30°) θ + 2θ − 30° = 90° 3θ − 30° = 90° 3θ = 120° ⇒ θ = 40°

50

33.

34.

35.

36.

37.

Chapter 2 Acute Angles and Right Triangles

sin ( 2θ + 10°) = cos (3θ − 20°) (2θ + 10°) + (3θ − 20°) = 90° 5θ − 10° = 90° 5θ = 100° ⇒ θ = 20° sec ( β + 10°) = csc ( 2 β + 20°) β + 10° ( ) + (2β + 20°) = 90° 3β + 30° = 90° 3β = 60° ⇒ β = 20° tan (3β + 4°) = cot (5β − 10°) (3β + 4°) + (5β − 10°) = 90° 8 β − 6° = 90° 8 β = 96° ⇒ β = 12° cot (5θ + 2°) = tan ( 2θ + 4°) (5θ + 2°) + (2θ + 4°) = 90° 7θ + 6° = 90° 7θ = 84° ⇒ θ = 12° sin (θ − 20°) = cos (2θ (θ − 20°) + (2θ + 5°) = 90°

+ 5°)

3θ − 15° = 90° 3θ = 105° ⇒ θ = 35°

38.

39.

40.

cos ( 2θ + 50°) = sin (2θ − 20°) (2θ + 50°) + (2θ − 20°) = 90° 4θ + 30° = 90° 4θ = 60° ⇒ θ = 15° sec (3β + 10°) = csc ( β + 8°) (3β + 10°) + ( β + 8°) = 90° 4 β + 18° = 90° 4 β = 72° ⇒ β = 18° csc ( β + 40°) = sec ( β − 20°) ( β + 40°) + ( β − 20°) = 90° 2 β + 20° = 90° 2 β = 70° ⇒ β = 35°

43. sin 46° < cos 46° Using the cofunction identity, cos 46° = sin (90° − 46°) = sin 44° . In the interval from 0º to 90º, as the angle increases, so does the sine of the angle, so sin 46° < sin 44° ⇒ sin 46° < cos 46° is false. 44. cos 28° < sin 28° Using the cofunction identity, sin 28° = cos (90° − 28° ) = cos 62° . In the interval from 0º to 90º, as the angle increases, the cosine of the angle decreases, so cos 28° < cos 62° ⇒ cos 28° < sin 28° is false. 45. tan 41° < cot 41° Using the cofunction identity, cot 41° = tan (90° − 41°) = tan 49° . In the interval from 0º to 90º, as the angle increases, the tangent of the angle increases, so tan 41° < tan 49° ⇒ tan 41° < cot 41° is true. 46. cot 30° < tan 40° Using the cofunction identity, cot 30° = tan (90° − 30°) = tan 60° . In the interval from 0º to 90º, as the angle increases, the tangent of the angle increases, so tan 60° < tan 40° ⇒ cot 30° < cot 40° is false. 47. sec 60° > sec 30° In the interval from 0º to 90º, as the angle increases, the cosine of the angle decreases, so the secant of the angle increases. Thus, sec 60° > sec 30° is true. 48. csc 20° < csc 30° In the interval from 0º to 90º, as the angle increases, sine of the angle increases, so cosecant of the angle decreases. Thus csc 20° < csc 30° is false. Use the following figures for exercises 49−64.

41. sin 50° > sin 40° In the interval from 0º to 90º, as the angle increases, so does the sine of the angle, so sin 50º > sin 40º is true. 42. tan 28° ≤ tan 40° In the interval from 0º to 90º, as the angle increases, the tangent of the angle increases, so tan 40º > tan 28º ⇒ tan 28° ≤ tan 40° is true.

side opposite 1 = side adjacent 3 1 3 3 = ⋅ = 3 3 3

49. tan 30° =

Section 2.1 Trigonometric Functions of Acute Angles

50. cot 30° = side adjacent = 3 = 3 side opposite 1

65.

51. sin 30° = side opposite = 1 hypotenuse 2 52. cos 30° = side adjacent = 3 hypotenuse 2 53.

hypotenuse 2 sec 30° = = side adjacent 3 2 3 2 3 = ⋅ = 3 3 3

66. sin 45° =

y 2 ⇒ y = 4 sin 45° = 4 ⋅ =2 2 4 2

and cos 45° =

x 4

⇒ x = 4 cos 45° = 4 ⋅

2 2

=2 2

hypotenuse 2 = =2 side opposite 1

54. csc 30° =

55. csc 45° = hypotenuse = 2 = 2 side opposite 1 56. sec 45° = hypotenuse = 2 = 2 side adjacent 1 57.

side adjacent 1 = hypotenuse 2 1 2 2 = ⋅ = 2 2 2

cos 45° =

58. cot 45° =

67. The legs of the right triangle provide the coordinates of P, 2 2 , 2 2 .

(

)

68.

side adjacent 1 = =1 side opposite 1

59. tan 45° = side opposite = 1 = 1 side adjacent 1 60. sin 45° = side opposite = 1 hypotenuse 2 1 2 2 = ⋅ = 2 2 2

y 3 ⇒ y = 2 sin 60° = 2 ⋅ = 3 2 2 x 1 and cos 60° = ⇒ x = 2 cos 60° = 2 ⋅ = 1 2 2 The legs of the right triangle provide the

61. sin 60° = side opposite = 3 hypotenuse 2

coordinates of P. P is 1, 3 .

62. cos 60° =

side adjacent 1 = hypotenuse 2

63. tan 60° =

side o pposite 3 = = 3 side djacent a 1

64.

hypotenuse 2 = side opposite 3 2 3 2 3 = ⋅ = 3 3 3

csc 60° =

sin 60° =

(

)

69. Y1 is sin x and Y2 is tan x. sin 0° = 0 tan 0° = 0 sin 30° = 0.5 tan 30° ≈ 0.57735 sin 45° ≈ 0.70711 tan 45° = 1 sin 60° ≈ 0.86603 tan 60° = 1.7321 sin 90° = 1 tan 90° : undefined

51

52

Chapter 2 Acute Angles and Right Triangles

75.

70. Y1 is cos x and Y2 is csc x. cos 0° = 1 csc 0° : undefined cos 30° ≈ 0.86603 csc 30° = 2 cos 45° ≈ 0.70711 csc 45° ≈ 1.4142 cos 60° = 0.5 csc 60° ≈ 1.1547 cos 90° = 0 csc 90° = 1 71. Since sin 60° = 90°, A = 60°.

3 and 60° is between 0° and 2

72. 0.7071067812 is a rational approximation for the exact value

2 (an irrational value). 2

73. The point of intersection is (0.70710678, 0.70710678). This corresponds to the point ⎛ 2 2⎞ ⎜⎝ 2 , 2 ⎟⎠ .

These coordinates are the sine and cosine of 45°. 74.

The line passes through (0, 0) and

(

)

3, 1 .

The slope is change in y over the change in x. 1 1 3 3 Thus, m = = ⋅ = and the 3 3 3 3 equation of the line is y =

3 x. 3

3 x, is the origin 3 (0, 0 ) . Let ( x, y ) be any other point on this line. Then, by the definition of slope, y −0 y 3 m= = = , but also, by the x−0 x 3 3 definition of tangent, tan θ = . Because 3 3 3 tan 30° = , the line y = x makes a 30° 3 3 angle with the positive x-axis. (See Exercise 75). 77. One point on the line y = 3x is the origin

76. One point on the line y =

(0, 0 ) . Let ( x, y ) be any other point on this

line. Then, by the definition of slope, y −0 y m= = = 3, but also, by the x−0 x definition of tangent, tan θ = 3. Because tan 60° = 3, the line y = 3x makes a 60° angle with the positive x-axis (See exercise 74).

(

)

The line passes through ( 0, 0) and 1, 3 . The slope is change in y over the change in x. 3 Thus, m = = 3 and the equation of the 1 line is y = 3 x.

Section 2.1 Trigonometric Functions of Acute Angles

78. (a) The diagonal forms two isosceles right triangles. Each angle formed by a side of the square and the diagonal measures 45º.

53

(c) Let x = the length of the perpendicular. Then apply the Pythagorean theorem. x 2 + k 2 = ( 2k ) ⇒ x 2 + k 2 = 4k 2 ⇒ x 2 = 3k 2 ⇒ x = 3k 2

The length of the perpendicular is

3k .

(d) In a 30º-60º right triangle, the hypotenuse is always 2 times as long as the shorter leg, and the longer leg has a length that is 3 times as long as that of the shorter (b) By the Pythagorean theorem, k 2 + k 2 = c 2 ⇒ 2k 2 = c 2 ⇒ c = Thus, the length of the diagonal is

2k . 2k .

leg. Also, the shorter leg is opposite the 30º angle, and the longer leg is opposite the 60º angle. 80. Apply the relationships between the lengths of the sides of a 30° − 60° right triangle first to the triangle on the left to find the values of a and b. In the 30° − 60° right triangle, the side opposite the 30° angle is 12 the length of the hypotenuse. The longer leg is shorter leg.

3 times the

(c) In a 45º-45º right triangle, the hypotenuse has a length that is 2 times as long as either leg. 79. (a) Each of the angles of the equilateral triangle has measure 13 (180°) = 60° .

1 (24) = 12 and b = a 3 = 12 3 2 Apply the relationships between the lengths of the sides of a 45° − 45° right triangle next to the triangle on the right to find the values of d and c. In the 45° − 45° right triangle, the sides opposite the 45° angles measure the same. a=

The hypotenuse is

2 times the measure of a leg. d = b = 12 3 and

(

c = d 2 = 12 3 (b) The perpendicular bisects the opposite side so the length of each side opposite each 30º angle is k.

)( 2 ) = 12

6

81. Apply the relationships between the lengths of the sides of a 30° − 60° right triangle first to the triangle on the left to find the values of y and x, and then to the triangle on the right to find the values of z and w. In the 30° − 60° right triangle, the side opposite the 30° angle is

1 2

leg is

the length of the hypotenuse. The longer 3 times the shorter leg. (continued on next page)

54

Chapter 2 Acute Angles and Right Triangles

83. Apply the relationships between the lengths of the sides of a 45° − 45° right triangle to the triangle on the left to find the values of p and r. In the 45° − 45° right triangle, the sides opposite the 45° angles measure the same. The hypotenuse is 2 times the measure of a leg.

(continued)

Thus, we have 1 9 9 3 y = (9 ) = and x = y 3 = 2 2 2 9 y 9 3 3 3 y = z 3, so z = = 2 = = , 6 2 3 3 ⎛3 3⎞ and w = 2z, so w = 2 ⎜ ⎟= 3 3 ⎝ 2 ⎠ 82. Apply the relationships between the lengths of the sides of a 30° − 60° right triangle first to the triangle on the right to find the values of m and a. In the 30° − 60° right triangle, the side opposite the 60° angle is 3 times as long as the side opposite to the 30° angle. The length of the hypotenuse is 2 times as long as the shorter leg (opposite the 30° angle).

Thus, we have p = 15 and r = p 2 = 15 2 Apply the relationships between the lengths of the sides of a 30° − 60° right triangle next to the triangle on the right to find the values of q and t. In the 30° − 60° right triangle, the side opposite the 60° angle is 3 times as long as the side opposite to the 30° angle. The length of the hypotenuse is 2 times as long as the shorter leg (opposite the 30° angle). Thus, we have r = q 3 ⇒ q=

r 15 2 15 2 3 = = ⋅ = 5 6 and 3 3 3 3

(

)

t = 2q = 2 5 6 = 10 6

Thus, we have 7=m 3⇒ m=

7 3

=

7 3

⋅

3 3

=

7 3 and 3

⎛ 7 3 ⎞ 14 3 a = 2m ⇒ a = 2 ⎜ ⎟= 3 ⎝ 3 ⎠ Apply the relationships between the lengths of the sides of a 45° − 45° right triangle next to the triangle on the left to find the values of n and q. In the 45° − 45° right triangle, the sides opposite the 45° angles measure the same. The hypotenuse is 2 times the measure of a leg. Thus, we have n = a =

14 3 and 3

⎛ 14 3 ⎞ 14 6 q=n 2=⎜ ⎟ 2= 3 . 3 ⎝ ⎠

84. Let h be the height of the equilateral triangle. h bisects the base, s, and forms two 30°–60° right triangles.

The formula for the area of a triangle is 1 A = bh. In this triangle, b = s. The height h 2 of the triangle is the side opposite the 60° angle in either 30°–60° right triangle. The side s opposite the 30° angle is . The height 2 s s 3 So the area of the entire is 3 ⋅ . = 2 2 1 ⎛ s 3 ⎞ s2 3 triangle is A = s ⎜ = . 2 ⎝ 2 ⎟⎠ 4

Section 2.2 Trigonometric Functions of Non-Acute Angles

6. B; 480° − 360° = 120° and 180° − 120° = 60° (120° is in quadrant II)

1 bh, we have 2 1 1 2 s2 A = ⋅ s ⋅ s = s or A = . 2 2 2

85. Since A =

7. 2 is a good choice for r because in a 30° − 60° right triangle, the hypotenuse is twice the length of the shorter side (the side opposite to the 30° angle). By choosing 2, one avoids introducing a fraction (or decimal) when determining the length of the shorter side. Choosing any even positive integer for r would have this result; however, 2 is the most convenient value.

86. Yes, the third angle can be found by subtracting the given acute angle from 90°, and the remaining two sides can be found using a trigonometric function involving the known angle and side.

Section 2.2 Trigonometric Functions of Non-Acute Angles

8. Answers may vary. The reference angle for an angle θ in QIII is given by θ ′ = θ − 180°. The trigonometric functions of θ are as follows: sin θ = − sin θ ′ csc θ = − cscθ ′ cos θ = − cos θ ′ sec θ = − sec θ ′ tan θ = tan θ ′ cot θ = cot θ ′

1. C; 180° − 98° = 82° (98° is in quadrant II) 2. F; 212° − 180° = 32° (212° is in quadrant III) 3. A; −135° + 360° = 225° and 225° − 180° = 45° (225° is in quadrant III)

9. Answers may vary. Two coterminal angle have the same values for their trigonometric functions because the two angles have the same reference angle.

4. B; −60° + 360° = 300° and 360° − 300° = 60° (300° is in quadrant IV)

10. Answers may vary. In quadrant II, the sine function is positive while the cosine and tangent functions are negative.

5. D; 750° − 2 ⋅ 360° = 30° (30° is in quadrant I) sin θ

cos θ

tan θ

11. 30°

1 2

3 2

3 3

12. 45°

2 2

1

13. 60°

3 2

14. 120°

3 2

2 2 1 2 cos 120° = − cos 60° 1 =− 2

15. 135°

2 2

−

2 2

16. 150°

sin 150° = sin 30° 1 = 2

−

3 2

θ

55

3

− 3 tan 135° = − tan 45° = −1 −

3 3

cot θ

sec θ

csc θ

3

2 3 3

2

1

2

2

3 2 3 cot 120° = − cot 60° sec120° = − sec 60° 3 =− = −2 3 cot 135° = − cot 45° − 2 = −1 sec150° cot 150° = − sec 30° = − cot 30° 2 3 =− =− 3 3

2 3 3 2 3 3

2

2

56

Chapter 2 Acute Angles and Right Triangles

θ 17. 210°

18. 240°

sin θ −

−

1 2

3 2

cos θ cos 210° = − cos 30° 3 =− 2 −

1 2

tan θ 3 3 tan 240° = tan 60° = 3

19. To find the reference angle for 300°, sketch this angle in standard position.

cot θ

3 cot 240° = cot 60° 3 = 3

sec θ

csc θ

sec 210° = − sec 30° 2 3 =− 3

−2

−2

−

2 3 3

sin 315° = − sin 45° = −

2 2

2 2 tan 315° = − tan 45° = −1 cot 315° = − cot 45° = −1 sec 315° = sec 45° = 2 csc 315° = − csc 45° = − 2

cos 315° = cos 45° =

The reference angle is 360° − 300° = 60°. Since 300° lies in quadrant IV, the sine, tangent, cotangent, and cosecant are negative. 3 sin 300° = − sin 60° = − 2 1 cos 300° = cos 60° = 2 tan 300° = − tan 60° = − 3 3 cot 300° = − cot 60° = − 3 sec 300° = sec 60° = 2 2 3 csc 300° = − csc 60° = − 3 20. To find the reference angle for 315°, sketch this angle in standard position.

The reference angle is 360° − 315° = 45°. Since 315° lies in quadrant IV, the sine, tangent, cotangent, and cosecant are negative.

21. To find the reference angle for 405°, sketch this angle in standard position.

The reference angle for 405° is 405° − 360° = 45°. Because 405° lies in quadrant I, the values of all of its trigonometric functions will be positive, so these values will be identical to the trigonometric function values for 45°. See the Function Values of Special Angles table on page 50.) 2 sin 405° = sin 45° = 2 2 cos 405° = cos 45° = 2 tan 405° = tan 45° = 1 cot 405° = cot 45° = 1 sec 405° = sec 45° = 2 csc 405° = csc 45° = 2

Section 2.2 Trigonometric Functions of Non-Acute Angles

22. To find the reference angle for 420º, sketch this angle in standard position.

sec ( 80°) = − sec 60° = −2 2 3 csc ( 480°) = csc 60°= 3

57

24. To find the reference angle for 495º, sketch this angle in standard position.

The reference angle for 420º is 420º − 360º = 60º. Because 420º lies in quadrant I, the values of all of its trigonometric functions will be positive, so these values will be identical to the trigonometric function values for 60°. See the Function Values of Special Angles table on page 50.) sin ( 420°) = sin 60° = cos ( 420°) = cos 60°=

3 2

1 2 tan ( 420°) = tan 60° = 3

cot ( 420°) = cot 60° =

3 3

sec ( 420°) = sec 60° = 2 2 3 csc ( 420°) = csc 60°= 3

495º is coterminal with 495º − 360º = 135º. The reference angle is 180º − 135º = 45º. Since 495° lies in quadrant II, the cosine, tangent, cotangent, and secant are negative. 2 sin 495° = sin 45° = 2 2 cos 495° = − cos 45° = − 2 tan 495° = − tan 45° = −1 cot 495° = − cot 45° = −1 sec 495° = − sec 45° = − 2 csc 495° = csc 45° = 2 25. To find the reference angle for 570º sketch this angle in standard position.

23. To find the reference angle for 480º, sketch this angle in standard position.

480º is coterminal with 480º − 360º = 120º. The reference angle is 180º − 120º = 60º. Because 480º lies in quadrant II, the cosine, tangent, cotangent, and secant are negative. sin ( 480°) = sin 60° =

3 2

cos ( 480°) = − cos 60°= −

1 2

tan ( 480°) = − tan 60° = − 3 3 cot ( 480°) = − cot 60° = − 3

570° is coterminal with 570º − 360º = 210º. The reference angle is 210º − 180º = 30º. Since 570º lies in quadrant III, the sine, cosine, secant, and cosecant are negative. 1 sin 570° = − sin 30° = − 2 3 cos 570° = − cos 30° = − 2 3 tan 570° = tan 30° = 3 cot 570° = cot 30° = 3 2 3 sec 570° = − sec 30° = − 3 csc 570° = − csc 30° = −2

58

Chapter 2 Acute Angles and Right Triangles

26. To find the reference angle for 750°, sketch this angle in standard position.

sin ( 420°) = sin 60° = cos ( 420°) = cos 60°= tan ( 420°) cot ( 420°)

sec ( 420°) csc ( 420°) 750° is coterminal with 30° because 750° − 2 ⋅ 360° = 750° − 720° = 30°. Since 750° lies in quadrant I, the values of all of its trigonometric functions will be positive, so these values will be identical to the trigonometric function values for 30°. 1 sin 750° = sin 30° = 2 3 cos 750° = cos 30° = 2 3 tan 750° = tan 30° = 3 cot 750° = cot 30° = 3 2 3 sec 750° = sec 30° = 3 csc 750° = csc 30° = 2 27. 1305º is coterminal with 1305° − 3 ⋅ 360° = 1305° = 1080° = 225° . The reference angle is 225º − 180º = 45º. Since 1305º lies in quadrant III, the sine, cosine, and secant and cosecant are negative. 2 sin 1305° = − sin 45° = − 2 2 cos1305° = − cos 45° = − 2 tan 1305° = tan 45° = 1 cot 1305° = cot 45° = 1 sec1305° = − sec 45° = − 2 csc1305° = − csc 45° = − 2 28. 1500º is coterminal with 1500° − 4 ⋅ 360° = 1500° − 1440° = 60° . Because 1500º lies in quadrant I, the values of all of its trigonometric functions will be positive, so these values will be identical to the trigonometric function values for 60°.

3 2

1 2 = tan 60° = 3 3 = cot 60° = 3 = sec 60° = 2 2 3 = csc 60°= 3

29. To find the reference angle for −300°, sketch this angle in standard position.

The reference angle for −300° is −300° + 360° = 60°. Because −300° lies in quadrant I, the values of all of its trigonometric functions will be positive, so these values will be identical to the trigonometric function values for 60°. See the Function Values of Special Angles table on page 50.) sin ( − 300°) = sin 60° = cos ( − 300°) = cos 60°= tan ( − 300°) cot ( − 300°)

sec ( − 300°) csc ( − 300°)

3 2

1 2 = tan 60° = 3 3 = cot 60° = 3 = sec 60° = 2 2 3 = csc 60°= 3

Section 2.2 Trigonometric Functions of Non-Acute Angles

59

30. −390° is coterminal with −390° + 2 ⋅ 360° = −390° + 720° = 330°. The reference angle is 360° − 330° = 30°. Since −390° lies in quadrant IV, the sine, tangent, cotangent, and cosecant are negative. 1 sin (−390°) = − sin 30° = − 2 3 cos (−390°) = cos 30° = 2 3 tan (−390°) = − tan 30° = − 3 cot (−390°) = − cot 30° = − 3 2 3 sec ( −390°) = sec 30° = 3 csc (−390°) = − csc 30° = −2

33. −1290° is coterminal with −1290° + 4 ⋅ 360° = −1290° + 1440° = 150°. The reference angle is 180° − 150° = 30°. Since −1290° lies in quadrant II, the cosine, tangent, cotangent, and secant are negative. 1 sin 2670° = sin 30° = 2 3 cos 2670° = − cos 30° = − 2 3 tan 2670° = − tan 30° = − 3

31. –510° is coterminal with −510° + 2 ⋅ 360° = −510° + 720° = 210°. The reference angle is 210° − 180° = 30°. Since –510° lies in quadrant III, the sine, cosine, and secant and cosecant are negative. 1 sin (−510°) = − sin 30° = − 2 3 cos (−510°) = − cos 30° = − 2 3 tan ( −510°) = tan 30° = 3 cot (−510°) = cot 30° = 3

34. −855º is coterminal with −855° + 3 ⋅ 360° = −855° + 1080° = 225° . The reference angle is 225º − 180º = 45º. Since −855º lies in quadrant III, the sine, cosine, and secant and cosecant are negative. 2 sin −855° = − sin 45° = −

sec ( −510°) = − sec 30° = −

2 3 3 csc (−510°) = − csc 30° = −2 32. −1020º is coterminal with −1020° + 3 ⋅ 360° = −1020° + 1080° = 60° . Because −1020º lies in quadrant I, the values of all of its trigonometric functions will be positive, so these values will be identical to the trigonometric function values for 60°. 3 sin ( 420°) = sin 60° = 2 1 cos ( 420°) = cos 60°= 2 tan ( 420°) = tan 60° = 3 3 cot ( 420°) = cot 60° = 3 sec ( 420°) = sec 60° = 2 2 3 csc ( 420°) = csc 60°= 3

cot 2670° = − cot 30° = − 3 2 3 sec 2670° = − sec 30° = − 3 csc 2670° = csc 30° = 2

(

)

2 2 cos (−855°) = − cos 45° = − 2 tan ( −855°) = tan 45° = 1 cot ( −855°) = cot 45° = 1 sec ( −855°) = − sec 45° = − 2 csc (−855°) = − csc 45° = − 2

35. −1860º is coterminal with −1860° + 6 ⋅ 360° = −1860° + 2160° = 300° . The reference angle is 360º − 300º = 60º. Since −1860º lies in quadrant IV, the sine, tangent, cotangent, and cosecant are negative. 3 sin ( − 1860°) = − sin 60° = − 2 1 cos ( − 1860°) = cos 60°= 2 tan ( − 1860°) = − tan 60° = − 3 3 cot ( − 1860°) = − cot 60° = − 3 sec ( − 1860°) = sec 60° = 2 2 3 csc ( − 1860°) = − csc 60°= − 3

60

Chapter 2 Acute Angles and Right Triangles

36. −2205º is coterminal with −2205° + 7 ⋅ 360° = −2205° + 2520° = 315° . The reference angle is 360º − 315º = 45º. Since −2205º lies in quadrant IV, the sine, tangent, cotangent, and cosecant are negative. 2 sin ( − 2205°) = − sin 45° = − 2 2 cos ( − 2205°) = cos 45°= 2 tan ( − 2205°) = − tan 45° = −1 cot ( − 2205°) = − cot 45° = −1 sec ( − 2205°) = sec 45° = 2 csc ( − 2205°) = − csc 45°= − 2

37. Since 1305° is coterminal with an angle of 1305° − 3 ⋅ 360° = 1305° − 1080° = 225°, it lies in quadrant III. Its reference angle is 225° − 180° = 45°. Since the sine is negative in quadrant III, we have 2 sin 1305° = − sin 45° = − . 2 38. Since 1500° is coterminal with an angle of 1500° − 4 ⋅ 360° = 1500° − 1440° = 60°, it lies in quadrant I. Because 1500° lies in quadrant I, the values of all of its trigonometric functions will be positive, so sin 1500° = sin 60° =

3 . 2

39. Since −510° is coterminal with an angle of −510° + 2 ⋅ 360° = −510° + 720° = 210°, it lies in quadrant III. Its reference angle is 210° − 180° = 30°. Since the cosine is negative in quadrant III, we have 3 cos (−510°) = − cos 30° = − . 2 40. Since −1020° is coterminal with an angle of −1020° + 3 ⋅ 360° = −1020° + 1080° = 60°, it lies in quadrant I. Because −1020° lies in quadrant I, the values of all of its trigonometric functions will be positive, so tan ( −1020°) = tan 60° = 3. 41. Since −855º is coterminal with −855° + 3 ⋅ 360° = −855° + 1080° = 225° , it lies in quadrant III. Its reference angle is

42. Since −495º is coterminal with an angle of −495° + 2 ⋅ 360° = −495° + 720° = 225° , it lies in quadrant III. Its reference angle is 225° − 180° = 45°. Since the secant is negative in quadrant III, we have sec ( −495°) = − sec 45° = − 2. 43. Since 3015º is coterminal with 3015° − 8 ⋅ 360° = 3015° − 2880° = 135° , it lies in quadrant II. Its reference angle is 180º − 135º = 45º. Since the tangent is negative in quadrant II, we have tan 3015° = − tan 45° = −1. 44. Since 2280º is coterminal with 2280° − 6 ⋅ 360° = 2280° − 2160° = 120° , it lies in quadrant II. Its reference angle is 180º − 120º = 60º. Since the cotangent is negative in quadrant II, we have 3 cot 2280° = − cot 60° = − . 3 2

2

⎛ 3⎞ ⎛ 1⎞ 45. sin 2 120° + cos 2 120° = ⎜ ⎟ + 2 ⎝ 2 ⎠ 3 1 = + =1 4 4 2 ⎛ 2⎞ ⎛ 2 2 2⎞ sin 225° + cos 225° = − 46. ⎜ ⎟ + ⎜− ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ 2 2 = + =1 4 4 2

2 2 2 47. 2 tan 120° + 3 sin 150° − cos 180° 2 2 ⎛ 1⎞ 2 = 2 − 3 +3 − −1 ⎜⎝ ⎟⎠ ( ) 2 ⎛ 1⎞ 23

(

)

= 2 (3) + 3 ⎜ ⎟ − 1 = ⎝4⎠ 4 48. cot 2 135° − sin 30° + 4 tan 45° 1 1 9 2 = (−1) − + 4 (1) = 1 − + 4 = 2 2 2 49. sin 2 225° − cos 2 270° + tan2 60° ⎛ = ⎜− 50.

⎝

2

2⎞ + 02 + 2 ⎟

( 3)

2

=

2 7 +3= 4 2

⎠

cot 2 90° − sec2 180° + csc2 135° 225º − 180º = 45º. Since the cosecant is negative in quadrant III, we have. csc (−855°) = − csc 45° = − 2.

= 0 2 − (−1) + 2

( 2)

2

= −1 + 2 = 1

Section 2.2 Trigonometric Functions of Non-Acute Angles

51. cos 2 60° + sec 2 150° − csc 2 210° 2 2 2 ⎛ 1⎞ ⎛ 2 3⎞ + − − (−2) = ⎜ 2 3 ⎟⎠ ⎝ 1 4 29 = + −4=− 4 3 12

?

57. sin 2 45° + cos 2 45° =1 ⎛ 2⎞ ⎛ 2⎞ 2 2 ⎜ 2 ⎟ + ⎜ 2 ⎟ = 4 + 4 =1 ⎝ ⎠ ⎝ ⎠ Since 1 = 1, the statement is true. 2

2

( 3)

4

− 04 = 1 + 9 = 10

53. cos (30° + 60°) = cos 30° + cos 60° Evaluate each side to determine whether this statement is true or false. cos (30° + 60°) = cos 90° = 0 and

tan 2 60° + 1 =

?

cos 30° + cos 60° = Since 0 ≠

3 +1

3 1 + = 2 2

2

58. tan 2 60° + 1 = sec 2 60° Evaluate each side to determine whether this statement is true or false.

2 4 4 52. cot 135° + tan 60° − sin 180°

= (−1) +

3 +1 2

+ 1 = 3 + 1 = 4 and

2

?

, the statement is false.

54. sin 30° + sin 60° = sin (30° + 60°) Evaluate each side to determine whether this statement is true or false. 1 3 1+ 3 + = 2 2 2 sin (30° + 60°) = sin 90° = 1 sin 30° + sin 60° =

and

1+ 3 ≠ 1, the given statement is false. 2

cos 90

Since 0 ≠ 2 , the statement is false. ?

60. sin ( 2 ⋅ 30°) = 2 sin 30° ⋅ cos 30° Evaluate each side to determine whether this statement is true or false. 3 and sin 2 ⋅ 30° = sin 60° =

(

)

2 sin 30° ⋅ cos 30° = 2

?

55. cos 60° = 2 cos 30° Evaluate each side to determine whether this statement is true or false. ⎛ 3⎞ 1 cos 60° = and 2cos 30° = 2 ⎜ ⎟= 3 2 ⎝ 2 ⎠

Since

56. cos 60° = 2 cos 30° − 1 Evaluate each side to determine whether this statement is true or false. 1 cos 60° = and 2 ⎛ 3⎞ ⎛ 3⎞ 2 cos 30° − 1 = 2 ⎜ ⎟ −1= 2 4 −1 2 ⎝ ⎠ 3 1 = −1 = 2 2 1 1 Since = , the statement is true. 2 2 2

3 ⎛ 1⎞⎛ 3 ⎞ = 2 ⎝⎜ 2 ⎠⎟ 2

3 3 , the statement is true. = 2 2

x 3 ⇒ x = r cos 30° = 6 ⋅ =3 3 r 2 1 °= ⋅ = °= ⇒ y=r and sin 30 sin 30 6 3 r 2 Since 150º is in quadrant II, the x- coordinate will be negative. The coordinates of P are cos 30° =

y 2

2

61. 150º is in quadrant II, so the reference angle is 180º − 150º = 30º.

1 ≠ 3, the statement is false. 2 ?

0 and

⎛ 2⎞ 2 cos 45° = 2 ⎜ ⎟= 2 ⎝ 2 ⎠

?

2

2

59. cos ( 2 ⋅ 45) ° = 2 cos 45° Evaluate each side to determine whether this statement is true or false. °= ( ⋅ )° =

2

Since

( 3)

sec 60° = 2 = 4 . Since 4 = 4, the statement is true. 2

cos 2 45

Since

61

(−3

)

3, 3 .

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