Exercises: 5.1 Algebra with Trigonometric Ratios

Bimal Kunwor; Donna Densmore; Jared Eusea; Yi Zhen

Chapter 5: Equations and Identities

Exercises: 5.1 Algebra with Trigonometric Ratios

Skills

Practice each skill in the Homework Problems listed:

Evaluate trigonometric expressions
Simplify trigonometric expressions
Recognize equivalent expressions
Multiply or expand trigonometric expressions
Factor trigonometric expressions

Exercises Homework 5.1

Exercise Group

For Problems 1–8, evaluate the expressions using exact values for the trigonometric ratios.

1.

$5 \tan 135 ° + 6 \cos 60 °$

2.

$3 \tan 240 ° + 8 \sin 300 °$

3.

$\sin (15 ° + 30 °)$

4.

$\cos (2 \cdot 75 °)$

5.

$8 \cos^{2} (30 °)$

6.

$12 \sin^{2} (315 °)$

7.

$3 \tan^{2} (150 °) - \sin^{2} (45 °)$

8.

$1 + \tan^{2} (120 °)$

Exercise Group

For Problems 9–16, evaluate the expressions for $x = 30 °, y = 45 °,$ and $z = 60 ° .$ Give exact values for your answers.

9.

$3 \sin x + 5 \cos y$

10.

$4 \tan y + 6 \cos y$

11.

$- 2 \tan 3 y$

12.

$\sin (3 z - 2 x)$

13.

$\cos^{2} x + \sin^{2} x$

14.

$7 \sin^{2} y + 7 \cos^{2} y$

15.

$\cos x \cos x - \sin x \sin z$

16.

$\tan (180 ° - x) \tan x$

Exercise Group

For Problems 17–22, evaluate the expressions using a calculator.

17.

$\sin (10 ° + 40 °)$
$\sin 10 ° + \sin 40 °$
$\sin 10 ° \cos 40 ° + \cos 10 ° \sin 40 °$

18.

$\cos (20 ° + 50 °)$
$\cos 20 ° + \cos 50 °$
$\cos 20 ° \cos 50 ° - \sin 20 ° \sin 50 °$

19.

$\cos (2 \cdot 24 °)$
$2 \cos 24 °$
$2 \cos^{2} (24 °) - 1$

20.

$\cos (3 \cdot 49 °)$
$3 \cos 49 °$
$4 \cos^{3} (49 °) - 3 \cos 49 °$

21.

$\cos^{2} (17 °) + \sin^{2} (17 °)$
$\cos^{2} (86 °) + \sin^{2} (86 °)$
$\cos^{2} (111 °) + \sin^{2} (111 °)$

22.

$\frac{1}{\cos^{2} (25 °)} - \tan^{2} (25 °)$
$\frac{1}{\cos^{2} (100 °)} - \tan^{2} (100 °)$
$\frac{1}{\cos^{2} (8 °)} - \tan^{2} (8 °)$

Exercise Group

For Problems 23–28, combine like terms.

23.

$3 x^{2} - x - 5 x^{2}$
$3 \cos^{2} θ - \cos θ - 5 \cos^{2} θ$

24.

$4 x + 5 y - y$
$4 \cos θ + 5 \sin θ - \sin θ$

25.

$- 3 S C + 7 S C$
$- 3 \sin θ \cos θ + 7 \sin θ \cos θ$

26.

$4 S C + 11 S C - 17 S C$
$4 \sin θ \cos θ + 11 \sin θ \cos θ - 17 \sin θ \cos θ$

27.

$- C^{2} S^{3} + 6 C^{2} S^{3}$
$- \cos^{2} θ \sin^{3} θ + 6 \cos^{2} θ \sin^{3} θ$

28.

$7 C^{2} S^{2} - (2 C S)^{2}$
$7 \cos^{2} θ \sin^{2} θ - (2 \cos θ \sin θ)^{2}$

Exercise Group

For Problems 29–34, simplify the expression, then evaluate.

29.

$\cos t + 2 \cos t \sin t - 3 \cos t,$ for $t = 143 °$

30.

$11 \tan w - 4 \tan w + 6 \tan w \cos w,$ for $w = 8 °$

31.

$7 \tan θ - 4 \tan ϕ + 3 \tan ϕ - 6 \tan θ,$ for $θ = 21 °, ϕ = 89 °$

32.

$5 \sin A + \sin B - 6 \sin B + 6 \sin A,$ for $A = 111 °, B = 26 °$

33.

$- \sin x \cos x - \sin (2 x) + 3 \sin x \cos x,$ for $x = 107 °$

34.

$4 \cos u \sin u + 3 \sin 2 u - 10 \sin u \cos u,$ for $u = 2 °$

Exercise Group

For Problems 35–44, decide whether or not the expressions are equivalent. Explain.

35.

$\cos (x + y), \cos x + \cos y$

36.

$\sin (θ - ϕ), \sin θ - \sin ϕ$

37.

$\tan (2 A), 2 \tan A$

38.

$\cos (\frac{1}{2} β), \frac{1}{2} \cos β$

39.

$(\sin α)^{2}, \sin^{2} α$

40.

$(\tan B)^{2}, \tan (B^{2})$

41.

$\sin 3 t + \sin 5 t, \sin 8 t$

42.

$3 \cos 2 x + 5 \cos 2 x, 8 \cos 2 x$

43.

$\tan (θ + 45 °) - \tan θ, \tan 45 °$

44.

$\sin (30 ° + z) + \sin (30 ° - z), \sin 60 °$

Exercise Group

For Problems 45–56, multiply or expand.

45.

$x (2 x - 1)$
$\sin A (2 \sin A - 1)$

46.

$q (5 r - q)$
$\cos θ (5 \sin θ - \cos θ)$

47.

$a (b - 3 a)$
$\tan A (\tan B - 3 \tan A)$

48.

$3 w (2 w - z)$
$3 \sin α (2 \sin α - \sin β)$

49.

$(C + 1) (2 C - 1)$
$(\cos ϕ + 1) (2 \cos ϕ - 1)$

50.

$(3 S - 2) (S + 1)$
$(3 \sin B - 2) (\sin B + 1)$

51.

$(a + b) (a - b)$
$(\cos θ + \cos ϕ) (\cos θ - \cos ϕ)$

52.

$(t - w) (t + 4 w)$
$(\tan α - \tan β) (\tan α + 4 \tan β)$

53.

$(1 - T)^{2}$
$(1 - \tan θ)^{2}$

54.

$(2 + 3 S)^{2}$
$(2 + 3 \sin θ)^{2}$

55.

$(T^{2} + 2) (T^{2} - 2)$
$(\tan^{2} θ + 2) (\tan^{2} θ - 2)$

56.

$(2 c^{2} - 3) (2 c^{2} + 3)$
$(2 \cos^{2} ϕ - 3) (2 \cos^{2} ϕ + 3)$

Exercise Group

For Problems 57–70, factor.

57.

$9 m + 15 n$
$9 \cos α + 15 \cos β$

58.

$12 p - 20 q$
$12 \sin θ - 20 \sin ϕ$

59.

$5 r^{2} - 10 q r$
$5 \tan^{2} C - 10 \tan B \tan C$

60.

$2 x^{2} - 6 x y$
$2 \sin^{2} a - 6 \sin A \cos A$

61.

$9 C^{2} - 1$
$9 \cos^{2} β - 1$

62.

$25 S^{2} - 16 S$
$25 \sin^{2} β - 16 \sin β$

63.

$6 T^{3} - 8 T^{2}$
$6 \tan^{3} A - 8 \tan^{2} A$

64.

$9 T^{2} - 15 T^{3}$
$9 \tan^{2} B - 15 \tan^{3} B$

65.

$t^{2} - t - 20$
$\tan^{2} θ - \tan θ - 20$

66.

$s^{2} + 5 s - 6$
$\sin^{2} A + 5 \sin A - 6$

67.

$3 c^{2} + 2 c - 1$
$3 \cos^{2} B + 2 \cos B - 1$

68.

$8 S^{2} - 6 S + 1$
$8 \sin^{2} ϕ - 6 \sin ϕ + 1$

69.

$6 S^{2} - 5 S - 1$
$6 \sin^{2} α - 5 \sin α - 1$

70.

$T^{2} - 4 T - 12$
$\tan^{2} α - 4 \tan α - 12$

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