Exercises: 8.1 Sum and Difference Formulas

1.

[latex]\sin \left(\beta + \dfrac{\pi}{4}\right)=\sin \beta + \dfrac{1}{\sqrt{2}}[/latex]

2.

[latex]\cos \left(\dfrac{\pi}{3} - t\right)=\dfrac{1}{2}-\cos t[/latex]

3.

[latex]\tan (z-w)=\dfrac{\sin (z-w)}{\cos (z-w)}[/latex]

4.

[latex]\sin 2\phi = 1 - \cos 2\phi[/latex]

5.

[latex]\sin \left(\dfrac{\pi}{2}-x\right)=1-\sin x[/latex]

6.

[latex]\sin (\pi - x)=\sin x[/latex]

7.

[latex]\cos^2 \alpha - \sin^2 \alpha = -1[/latex]

8.

[latex]\tan^{-1} s = \dfrac{1}{\tan s}[/latex]

9.

If [latex]\sin x = -0.4[/latex] and [latex]\cos x \gt 0,[/latex] find an exact value for [latex]\cos \left(x + \dfrac{3\pi}{4}\right)\text{.}[/latex]

10.

If [latex]\cos x = -0.75[/latex] and [latex]\sin x \lt 0,[/latex] find an exact value for [latex]\cos \left(x - \dfrac{4\pi}{3}\right)\text{.}[/latex]

11.

If [latex]\cos \theta = \dfrac{-3}{8},~ \pi \lt \theta \lt \dfrac{3\pi}{2},[/latex] and [latex]\sin \phi = \dfrac{1}{4},~ \dfrac{\pi}{2} \lt \phi \lt \pi,[/latex] find exact values for

1. [latex]\displaystyle \sin (\theta + \phi)[/latex]
2. [latex]\displaystyle \tan (\theta + \phi)[/latex]

12.

If [latex]\sin \rho = \dfrac{5}{6},~ \dfrac{\pi}{2} \lt \rho \lt \pi,[/latex] and [latex]\cos \mu = \dfrac{-1}{3},~ \dfrac{\pi}{2} \lt \mu \lt \pi,[/latex] find exact values for

1. [latex]\displaystyle \cos (\rho - \mu)[/latex]
2. [latex]\displaystyle \tan (\rho - \mu)[/latex]

13.

If [latex]\tan (x + y) =2[/latex] and [latex]\tan y = \dfrac{1}{3},[/latex] find [latex]\tan x\text{.}[/latex]

14.

If [latex]\tan (x - y) =\dfrac{1}{4}[/latex] and [latex]\tan x = 4,[/latex] find [latex]\tan y\text{.}[/latex]

15.

[latex]\tan \left(t - \dfrac{5\pi}{3}\right)[/latex]

16.

[latex]\cos \left(s+ \dfrac{7\pi}{4}\right)[/latex]

17.

1. [latex]\displaystyle \sin \theta[/latex]
2. [latex]\displaystyle \cos \theta[/latex]
3. [latex]\displaystyle \tan \theta[/latex]
4. [latex]\displaystyle \sin 2\theta[/latex]
5. [latex]\displaystyle \cos 2\theta[/latex]
6. [latex]\displaystyle \tan 2\theta[/latex]

18.

1. [latex]\displaystyle \sin \phi[/latex]
2. [latex]\displaystyle \cos \phi[/latex]
3. [latex]\displaystyle \tan \phi[/latex]
4. [latex]\displaystyle \sin 2\phi[/latex]
5. [latex]\displaystyle \cos 2\phi[/latex]
6. [latex]\displaystyle \tan 2\phi[/latex]

19.

[latex]\sin 4x \cos 5x + \cos 4x \sin 5x[/latex]

20.

[latex]\cos 3\beta \cos 1.5 - \sin 3\beta \sin 1.5[/latex]

21.

[latex]\dfrac{\tan 2\phi - \tan 2}{1 + \tan 2\phi \tan 2}[/latex]

22.

[latex]\dfrac{\tan \dfrac{5\pi}{9} - \tan \dfrac{2\pi}{9}}{1 + \tan \dfrac{5\pi}{9} \tan \dfrac{2\pi}{9}}[/latex]

23.

[latex]2\sin 4\theta \cos 4\theta[/latex]

24.

[latex]1-2\sin^2 3\phi[/latex]

25.

[latex]\cos 2\theta - \sin \theta = 1[/latex]

26.

[latex]\tan 2z + \tan z = 0[/latex]

27.

[latex]f(x) = 4x - x^3[/latex]

28.

[latex]g(x) = 5 + \sqrt[3]{x - 2}[/latex]

29.

1. [latex]\displaystyle \tan^{-1}(-\sqrt{3})[/latex]
2. [latex]\displaystyle \arccos \left(-\dfrac{1}{2}\right)[/latex]

30.

1. [latex]\displaystyle \arcsin (-1)[/latex]
2. [latex]\displaystyle \cos^{-1}(-1)[/latex]

31.

An IMAX movie screen is 52.8 feet high.

1. If your line of sight is level with the bottom of the screen, write an expression for the angle subtended by the screen when you sit [latex]x[/latex] feet away.
2. Evaluate your expression for [latex]x = 20[/latex] feet and for [latex]x = 100[/latex] feet.

32.

Rembrandt’s painting The Night Watch measures 13 feet high by 16 feet wide.

1. Write an expression for the angle subtended by the width of the painting if you sit [latex]d[/latex] feet back from the center of the painting.
2. Evaluate your expression for [latex]d = 10[/latex] feet and for [latex]d = 25[/latex] feet.

33.

[latex]v_y = v_0\sin \theta - gt[/latex]

34.

[latex]\Delta W = -q_0 E\cos (\pi - \theta)\Delta l[/latex]

35.

[latex]\cos\left[\tan^{-1}\left(\dfrac{-\sqrt{5}}{2}\right)\right][/latex]

36.

[latex]\tan\left[\sin^{-1}\left(\dfrac{2}{7}\right)\right][/latex]

37.

[latex]\sin(\cos^{-1}2t)[/latex]

38.

[latex]\tan(\cos^{-1}m)[/latex]

39.

Explain why one of the expressions [latex]\sin^{-1}x[/latex] or [latex]\sin^{-1}\left(\dfrac{1}{x}\right)[/latex] must be undefined.

40.

Does [latex]\sin^{-1}(-x) = -\sin^{-1}(x)\text{?}[/latex] Does [latex]\cos^{-1}(-x) = -\cos^{-1}(x)\text{?}[/latex]

41.

1. [latex]\displaystyle \csc 27°[/latex]
2. [latex]\displaystyle \sec 108°[/latex]
3. [latex]\displaystyle \cot 245°[/latex]

42.

1. [latex]\displaystyle \csc 5.3[/latex]
2. [latex]\displaystyle \cot 0.98[/latex]
3. [latex]\displaystyle \sec 2.17[/latex]

43.

44.

45.

46.

47.

48.

49.

[latex]6\cos \alpha = -5\text{,}[/latex] [latex]~ 180° \lt \alpha \lt 270°[/latex]

50.

[latex]4\sin \theta = 3,~ \theta[/latex] is obtuse

51.

52.

53.

54.

55.

[latex]2\sin \alpha - k = 0,~\dfrac{\pi}{2} \lt \alpha \lt \pi[/latex]

56.

[latex]h\cos \beta - 3 = 0,~\dfrac{3\pi}{2} \lt \beta \lt 2\pi[/latex]

57.

58.

59.

[latex]4\cot \dfrac{3\pi}{4} - \sec^2 \dfrac{\pi}{3}[/latex]

60.

[latex]\dfrac{1}{2}\csc \dfrac{2\pi}{3} + \tan^2 \dfrac{5\pi}{6}[/latex]

61.

[latex]\csc \dfrac{7\pi}{6}\cos \dfrac{5\pi}{4}[/latex]

62.

[latex]\sec \dfrac{7\pi}{4}\cot \dfrac{4\pi}{3}[/latex]

63.

[latex]3\csc \theta + 2 = 12[/latex]

64.

[latex]5\cot \theta + 15 = -3[/latex]

65.

The graph has vertical asymptotes at multiples of [latex]\pi\text{.}[/latex]

66.

The graph has a horizontal asymptote at [latex]\dfrac{\pi}{2}\text{.}[/latex]

67.

The function values are the reciprocals of [latex]y = \cos x\text{.}[/latex]

68.

The function is defined only for [latex]x[/latex]-values between [latex]-1[/latex] and [latex]1\text{,}[/latex] inclusive.

69.

None of the function values lie between [latex]-1[/latex] and [latex]1\text{.}[/latex]

70.

The graph includes the origin.

71.

[latex]f(x)=\tan x(\cos x - \cot x)[/latex]

72.

[latex]g(x)=\csc x - \cot x\cos x[/latex]

73.

[latex]G(x) = \sin x(\sec x - \csc x)[/latex]

74.

[latex]F(x) = \dfrac{1}{2}\left(\dfrac{\cos x}{1+\sin x} + \dfrac{1+ \sin x}{\cos x}\right)[/latex]

75.

[latex]1-\dfrac{\sin x}{\csc x}[/latex]

76.

[latex]\dfrac{\sin x}{\csc x}+\dfrac{\cos x}{\sec x}[/latex]

77.

[latex]\dfrac{2+\tan^2 B}{\sec^2 B} - 1[/latex]

78.

[latex]\dfrac{\csc t}{\tan t + \cot t}[/latex]

79.

[latex]\dfrac{\sqrt{16+x^2}}{x},~~x = 4\tan \theta[/latex]

80.

[latex]x\sqrt{4-x^2},~~x=2\sin \theta[/latex]

81.

[latex]\dfrac{x^2 - 3}{x},~~x=\sqrt{3}\sec \theta[/latex]

82.

[latex]\dfrac{x}{\sqrt{x^2+2}},~~x=\sqrt{2}\tan \theta[/latex]

83

This problem outlines a geometric proof of difference of angles formula for tangent.

1. In the figure below left, [latex]\alpha=\angle ABC[/latex] and [latex]\beta = \angle DBC\text{.}[/latex] Write expressions in terms of [latex]\alpha[/latex] and [latex]\beta[/latex] for the sides [latex]AC,~DC,[/latex] and [latex]AD\text{.}[/latex]
2. In the figure above right, explain why [latex]\triangle ABC[/latex] is similar to [latex]\triangle FBE\text{.}[/latex]
3. Explain why [latex]\angle FDC = \alpha\text{.}[/latex]
4. Write an expression in terms of [latex]\alpha[/latex] and [latex]\beta[/latex] for side [latex]CF\text{.}[/latex]
5. Explain why [latex]\triangle FBE[/latex] is similar to [latex]\triangle ADE\text{.}[/latex]
6. Justify each equality in the statement
   [latex]\tan (\alpha - \beta) = \dfrac{DE}{BE} = \dfrac{AD}{BF} = \dfrac{\tan \alpha - \tan \beta}{1+\tan \alpha \tan \beta}[/latex]

84.

Let [latex]L_1[/latex] and [latex]L_2[/latex] be two lines with slopes [latex]m_1[/latex] and [latex]m_2\text{,}[/latex] respectively, and let [latex]\theta[/latex] be the acute angle formed between the two lines. Use an identity to show that

[latex]\tan \theta = \dfrac{m_2-m_1}{1+m_1m_2}[/latex]

85.

In the figure above, find the diameter of the circumscribing circle, the angle [latex]\alpha\text{,}[/latex] and the sides [latex]a[/latex] and [latex]b\text{.}[/latex]

86.

A triangle has one side of length 17 cm and the angle opposite is [latex]26°\text{.}[/latex] Find the diameter of the circle that circumscribes the triangle.

1.

1. Sketch an angle [latex]\alpha[/latex] in standard position, with [latex]\dfrac{\pi}{2} \lt \alpha \lt \pi{.}[/latex] Also sketch the angle [latex]-\alpha{.}[/latex]
2. Choose a point on the terminal side of [latex]\alpha{,}[/latex] and show that the negative angle identities hold for [latex]\alpha{.}[/latex]

2.

1. Sketch an angle [latex]\beta[/latex] in standard position, with [latex]\pi \lt \beta \lt \dfrac{3\pi}{2}{.}[/latex] Also sketch the angle [latex]-\beta{.}[/latex]
2. Choose a point on the terminal side of [latex]\beta[/latex] and show that the negative angle identities hold for [latex]\beta{.}[/latex]

3.

Given that [latex]~~\sin \dfrac{7\pi}{12} = \dfrac{\sqrt{2} + \sqrt{6}}{4}~~{,}[/latex] find [latex]\sin \dfrac{-7\pi}{12}{.}[/latex] Sketch both angles.

4.

Given that [latex]~~\cos \dfrac{7\pi}{12} = \dfrac{\sqrt{2} - \sqrt{6}}{4}~~{,}[/latex] find [latex]\cos \dfrac{-7\pi}{12}{.}[/latex] Sketch both angles.

5.

If [latex]~~\cos(2x-0.3)=0.24~~[/latex] and [latex]~~\sin(2x-0.3) \lt 0~~{,}[/latex] find [latex]\cos(0.3-2x)[/latex] and [latex]\sin(0.3-2x){.}[/latex]

6.

If [latex]~~\sin(1.5-\phi)=-0.28~~[/latex] and [latex]~~\cos(1.5-\phi) \gt 0~~{,}[/latex] find [latex]\sin(\phi-1.5)[/latex] and [latex]\cos(\phi-1.5){.}[/latex]

7.

Show that [latex]\cos(45°+45°)[/latex] is not equal to [latex]\cos 45° + \cos 45°{.}[/latex]

8.

Show that [latex]\tan(60°-30°)[/latex] is not equal to [latex]\tan 60° - \tan 30°{.}[/latex]

9.

Use your calculator to verify that [latex]\tan (87°-29°)[/latex] is not equal to [latex]\tan 87° - \tan 29°{.}[/latex]

10.

Use your calculator to verify that [latex]\cos (52°+64°)[/latex] is not equal to [latex]\cos 52° + \cos 64°{.}[/latex]

11.

Use graphs to show that [latex]\sin\left(x-\dfrac{\pi}{6}\right)[/latex] is not equivalent to [latex]\sin x - \sin\dfrac{\pi}{6}{.}[/latex]

12.

Use graphs to show that [latex]\tan\left(x+\dfrac{\pi}{4}\right)[/latex] is not equivalent to [latex]\tan x + \tan\dfrac{\pi}{4}{.}[/latex]

13.

Suppose [latex]\cos \alpha = \dfrac{3}{5},~ \sin \alpha = \dfrac{4}{5},~ \cos \beta = \dfrac{5}{13}{,}[/latex] and [latex]\sin \beta = \dfrac{-12}{13}{.}[/latex] Evaluate the following.

14.

Suppose [latex]\cos \alpha = \dfrac{-2}{3},~ \sin \alpha = \dfrac{\sqrt{5}}{3},~ \cos \beta = \dfrac{\sqrt{3}}{2}{,}[/latex] and [latex]\sin \beta = \dfrac{-1}{2}{.}[/latex] Evaluate the following.

1. [latex]\displaystyle \cos(\alpha - \beta)[/latex]
2. [latex]\displaystyle \sin(\alpha - \beta)[/latex]
3. [latex]\displaystyle \tan(\alpha - \beta)[/latex]

15.

If [latex]\tan t = \dfrac{3}{4}[/latex] and [latex]\tan s = \dfrac{-7}{24}{,}[/latex] find exact values for:

1. [latex]\displaystyle \tan(s+t)[/latex]
2. [latex]\displaystyle \tan(s-t)[/latex]

16.

If [latex]\tan x = -3[/latex] and [latex]\tan y = -5{,}[/latex] find exact values for:

1. [latex]\displaystyle \tan(x+y)[/latex]
2. [latex]\displaystyle \tan(x-y)[/latex]

17.

Suppose [latex]\cos \theta = \dfrac{15}{17}[/latex] and [latex]\sin \phi = \dfrac{3}{5}{,}[/latex] where [latex]\theta[/latex] and [latex]\phi[/latex] are in quadrant I. Evaluate the following.

1. [latex]\displaystyle \cos(\theta + \phi)[/latex]
2. [latex]\displaystyle \tan(\theta - \phi)[/latex]

18.

Suppose [latex]\cos \theta = \dfrac{15}{17}{,}[/latex] where [latex]\theta[/latex] is in quadrant IV, and [latex]\sin \phi = \dfrac{3}{5}{,}[/latex] where [latex]\phi[/latex] is in quadrant II. Evaluate the following.

1. [latex]\displaystyle \sin(\theta - \phi)[/latex]
2. [latex]\displaystyle \tan(\theta + \phi)[/latex]

19.

If [latex]\sin \alpha = \dfrac{12}{13},~ \dfrac{\pi}{2} \lt\alpha \lt \pi{,}[/latex] and [latex]\cos \beta = \dfrac{-3}{5},~ \pi \lt \beta \lt \dfrac{3\pi}{2}{,}[/latex] find exact values for:

1. [latex]\displaystyle \sin(\alpha + \beta)[/latex]
2. [latex]\displaystyle \cos(\alpha + \beta)[/latex]
3. [latex]\displaystyle \tan(\alpha + \beta)[/latex]
4. Sketch the angles [latex]\alpha, \beta[/latex] and [latex]\alpha + \beta{.}[/latex]

20.

If [latex]\cos \alpha = \dfrac{3}{8},~ \dfrac{3\pi}{2} \lt\alpha \lt 2\pi{,}[/latex] and [latex]\sin \beta = \dfrac{-1}{4},~ \pi \lt \beta \lt \dfrac{3\pi}{2}{,}[/latex] find exact values for:

1. [latex]\displaystyle \sin(\alpha - \beta)[/latex]
2. [latex]\displaystyle \cos(\alpha - \beta)[/latex]
3. [latex]\displaystyle \tan(\alpha - \beta)[/latex]
4. Sketch the angles [latex]\alpha, \beta[/latex] and [latex]\alpha - \beta{.}[/latex]

21.

Find the exact values of [latex]\cos 15°[/latex] and [latex]\tan 15°{.}[/latex]

22.

Find the exact values of [latex]\sin 165°[/latex] and [latex]\tan 165°{.}[/latex]

23.

If [latex]\sin \theta = 0.2[/latex] and [latex]\cos \theta \gt 0{,}[/latex] find [latex]\sin \left(\theta + \dfrac{\pi}{3}\right){.}[/latex]

24.

If [latex]\cos \theta = 0.6[/latex] and [latex]\sin \theta \lt 0{,}[/latex] find [latex]\cos \left(\theta + \dfrac{3\pi}{4}\right){.}[/latex]

25.

[latex]\sin (\theta - 270°)[/latex]

26.

[latex]\cos(270° + \theta)[/latex]

27.

[latex]\cos\left(t + \dfrac{\pi}{6}\right)[/latex]

28.

[latex]\sin \left(t - \dfrac{2\pi}{3}\right)[/latex]

29.

[latex]\tan\left(\beta - \dfrac{\pi}{6}\right)[/latex]

30.

[latex]\tan \left(\phi + \dfrac{\pi}{4}\right)[/latex]

31.

Does [latex]\sin(2\cdot 80°) = 2\sin 80°{?}[/latex]

32.

Does [latex]\cos(2\cdot 25°) = 2\cos 25°{?}[/latex]

33.

Does [latex]\tan(2\cdot 70°) = 2\tan 70°{?}[/latex]

34.

Does [latex]\tan(2\cdot 100°) = 2\tan 100°{?}[/latex]

35.

[latex]\sin 90° = 2\sin 45° \cos 45°[/latex]

36.

[latex]\sin 60° = 2\sin 30° \cos 30°[/latex]

37.

[latex]\cos 60° = \cos^2 30° - \sin^2 30°[/latex]

38.

[latex]\tan 60° = \dfrac{2\tan 30°}{1 - \tan^2 30°}[/latex]

39.

If [latex]\cos \alpha = 0.32{,}[/latex] then [latex]\cos 2\alpha = 2(0.32) = 0.64{.}[/latex]

40.

If [latex]\cos 2\beta = 0.86{,}[/latex] then [latex]\cos \beta = 0.43{,}[/latex] so [latex]\beta = \cos^{-1}(0.43){.}[/latex]

41.

If [latex]\sin 2\theta = h{,}[/latex] then [latex]\sin \theta = \dfrac{h}{2}{,}[/latex] so [latex]\theta = \sin^{-1}\left(\dfrac{h}{2}\right){.}[/latex]

42.

If [latex]\cos \phi = r{,}[/latex] then [latex]\cos 2\phi = 2r{.}[/latex]

43.

[latex]2\sin 34° \cos 34°[/latex]

44.

[latex]\cos^2 \dfrac{\pi}{10} - \sin^2 \dfrac{\pi}{10}[/latex]

45.

[latex]1 - 2\sin^2 \dfrac{\pi}{16}[/latex]

46.

[latex]2\cos^2 18° - 1[/latex]

47.

[latex]\cos^2 3\theta - \sin^2 3\theta[/latex]

48.

[latex]2\sin 2\alpha \cos 2\alpha[/latex]

49.

[latex]2\sin 5t \cos 5t[/latex]

50.

[latex]\cos^2 4w - \sin^2 4w[/latex]

51.

[latex]\dfrac{2\tan 64°}{1- \tan^2 64°}[/latex]

52.

[latex]\dfrac{2\tan \dfrac{\pi}{3}}{1- \tan^2 \dfrac{\pi}{3}}[/latex]

53.

[latex]2\cos^2 2\beta - 1[/latex]

54.

[latex]1 - 2\sin^2 6s[/latex]

55.

1. [latex]\displaystyle \sin \alpha[/latex]

56.

1. [latex]\displaystyle \sin \beta[/latex]
2. [latex]\displaystyle \cos \beta[/latex]
3. [latex]\displaystyle \tan \beta[/latex]
4. [latex]\displaystyle \sin 2\beta[/latex]
5. [latex]\displaystyle \cos 2\beta[/latex]
6. [latex]\displaystyle \tan 2\beta[/latex]

57.

1. [latex]\displaystyle \sin s[/latex]
2. [latex]\displaystyle \cos s[/latex]
3. [latex]\displaystyle \tan s[/latex]
4. [latex]\displaystyle \sin s[/latex]
5. [latex]\displaystyle \cos s[/latex]
6. [latex]\displaystyle \tan s[/latex]

58.

1. [latex]\displaystyle \sin t[/latex]
2. [latex]\displaystyle \cos t[/latex]
3. [latex]\displaystyle \tan t[/latex]
4. [latex]\displaystyle \sin t[/latex]
5. [latex]\displaystyle \cos t[/latex]
6. [latex]\displaystyle \tan t[/latex]

59.

Suppose [latex]\cos \theta = \dfrac{12}{13}[/latex] and [latex]\dfrac{3\pi}{2} \lt \theta \lt 2\pi{.}[/latex] Compute exact values for:

1. [latex]\displaystyle \sin \theta[/latex]
2. [latex]\displaystyle \sin 2\theta[/latex]
3. [latex]\displaystyle \cos 2\theta[/latex]
4. [latex]\displaystyle \tan 2\theta[/latex]
5. Sketch the angles [latex]\theta[/latex] and [latex]2\theta{.}[/latex]

60.

Suppose [latex]\sin \phi = \dfrac{5}{6}[/latex] and [latex]\dfrac{\pi}{2} \lt \phi \lt \pi{.}[/latex] Compute exact values for:

1. [latex]\displaystyle \cos \phi[/latex]
2. [latex]\displaystyle \sin 2\phi[/latex]
3. [latex]\displaystyle \cos 2\phi[/latex]
4. [latex]\displaystyle \tan 2\phi[/latex]
5. Sketch the angles [latex]\phi[/latex] and [latex]2\phi{.}[/latex]

61.

If [latex]\tan u = -4[/latex] and [latex]270° \lt u \lt 360°{,}[/latex] find exact values for:

1. [latex]\displaystyle \tan 2u[/latex]
2. [latex]\displaystyle \cos 2u[/latex]
3. [latex]\displaystyle \sin 2u[/latex]

62.

If [latex]\tan v = \dfrac{2}{3}[/latex] and [latex]180° \lt u \lt 270°{,}[/latex] find exact values for:

1. [latex]\displaystyle \tan 2v[/latex]
2. [latex]\displaystyle \cos 2v[/latex]
3. [latex]\displaystyle \sin 2v[/latex]

63.

[latex]\sin 2\theta + \sqrt{2} \cos \theta = 0[/latex]

64.

[latex]\sin 2\alpha \sin \alpha =\cos \alpha[/latex]

65.

[latex]\cos 2t -5\cos t + 3 = 0[/latex]

66.

[latex]\cos 2x + 3\sin x = 2[/latex]

67.

[latex]\tan 2\beta + 2\sin \beta = 0[/latex]

68.

[latex]\tan 2z - 2\cos z = 0[/latex]

69.

[latex]3\cos \phi - \sin \left(\dfrac{\pi}{2} - \phi\right) = \sqrt{3}[/latex]

70.

[latex]\sin w + \cos\left(\dfrac{\pi}{2} - w\right) = 1[/latex]

71.

[latex]\sin 2\phi \cos \phi + \cos 2\phi \sin \phi = 1[/latex]

72.

[latex]\cos \theta \cos 3\theta + \sin \theta \sin 3\theta = \dfrac{\sqrt{2}}{2}[/latex]

73.

Use the sum of angles formulas for sine and cosine to derive a formula for each expression. Then use graphs to verify your formula.

1. [latex]\displaystyle \cos (\theta + 90°)[/latex]
2. [latex]\displaystyle \sin (\theta + 90°)[/latex]

74.

Use the sum of angles formulas for sine and cosine to derive a formula for each expression. Then use graphs to verify your formula.

1. [latex]\displaystyle \cos (\theta + \pi)[/latex]
2. [latex]\displaystyle \sin (\theta + \pi)[/latex]

75.

Use the difference of angles formulas for sine and cosine to prove that:

1. [latex]\displaystyle \cos \left(\dfrac{\pi}{2} -\theta\right) = \sin \theta[/latex]
2. [latex]\displaystyle \sin \left(\dfrac{\pi}{2} -\theta\right) = \cos \theta[/latex]

76.

Use the difference of angles formulas for sine and cosine to derive formulas for:

1. [latex]\displaystyle \cos \left(\theta - \dfrac{\pi}{2}\right)[/latex]
2. [latex]\displaystyle \sin \left(\theta - \dfrac{\pi}{2}\right)[/latex]

77.

Prove the double angle identity [latex]\sin 2\theta = 2\sin \theta \cos\theta{.}[/latex] (Hint: Start with the sum of angles formula for sine and replace both [latex]\alpha[/latex] and [latex]\beta[/latex] by [latex]\theta{.}[/latex])

78.

Prove the double angle identity [latex]\cos 2\theta = \cos^2 \theta - \sin^2 \theta{.}[/latex] (Hint: Start with the sum of angles formula for sine and replace both [latex]\alpha[/latex] and [latex]\beta[/latex] by [latex]\theta{.}[/latex])

79.

[latex]\sin \left(\dfrac{\pi}{2} + \beta\right) = 1 + \sin \beta[/latex]

80.

[latex]\cos \left(\dfrac{\pi}{3} - \beta\right) =\cos \left(\beta - \dfrac{\pi}{3}\right)[/latex]

81.

[latex]\sin(A+180°) = -\sin A[/latex]

82.

[latex]\tan \theta + \tan(-\theta) = 0[/latex]

83.

[latex]\cos 4\theta = 4\cos \theta[/latex]

84.

[latex]\cos\left(\phi + \dfrac{\pi}{3}\right)= \dfrac{1}{2} + \cos \phi[/latex]

85.

[latex]\sin\left(x + \dfrac{\pi}{4}\right) = \dfrac{\sqrt{2}}{2}(\sin x + \cos x)[/latex]

86.

[latex]2\cos\left(x - \dfrac{\pi}{6}\right) = \sin x + \sqrt{3}\cos x[/latex]

87.

[latex]\sin\left(x - \dfrac{\pi}{3}\right) + \cos \left(x + \dfrac{\pi}{6}\right)= 0[/latex]

88.

[latex]\cos 2x = (\cos x + \sin x)(\cos x - \sin x)[/latex]

89.

The figure below shows a right triangle inscribed in a rectangle.

90.

The figure below shows a right triangle inscribed in a rectangle.

1. Label the legs [latex]l_1[/latex] and [latex]l_2[/latex] of the right triangle with their lengths.
2. Explain why [latex]\theta_1 = \beta[/latex] and [latex]\theta_2 = \alpha -\beta{.}[/latex] Label the diagram with these angles.
3. Label the legs [latex]s_1[/latex] and [latex]s_2[/latex] of the bottom triangle with their lengths.
4. Label the legs [latex]s_3[/latex] and [latex]s_4[/latex] of the top left triangle with their lengths.
5. Label the legs [latex]s_5[/latex] and [latex]s_6[/latex] of the top right triangle with their lengths.
6. Use the fact that the opposite sides of a rectangle are equal to state the subtraction formulas for sine and cosine.

91.

Follow the steps to prove the difference of angles formula for cosine,
[latex]\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta[/latex]

1. Write an expression for [latex](AB)^2{,}[/latex] the square of the distance between the points [latex]A[/latex] and [latex]B{,}[/latex] using the law of cosines for [latex]\triangle AOB{.}[/latex]
2. Write another expression for [latex](AB)^2[/latex] using the distance formula and the coordinates of [latex]A[/latex] and [latex]B{.}[/latex]
3. Equate the two expressions for [latex](AB)^2[/latex] you obtained in (a) and (b). Simplify the equation to obtain [latex]\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta{.}[/latex]