Solutions for Double-Angle, Half-Angle, and Reduction Formulas

Solutions to Try Its

1. [latex]\cos \left(2\alpha \right)=\frac{7}{32}[/latex]

2. [latex]{\cos }^{4}\theta -{\sin }^{4}\theta =\left({\cos }^{2}\theta +{\sin }^{2}\theta \right)\left({\cos }^{2}\theta -{\sin }^{2}\theta \right)=\cos \left(2\theta \right)[/latex]

3. [latex]\cos \left(2\theta \right)\cos \theta =\left({\cos }^{2}\theta -{\sin }^{2}\theta \right)\cos \theta ={\cos }^{3}\theta -\cos \theta {\sin }^{2}\theta [/latex]

4. [latex]\begin{array}{ll}10{\cos }^{4}x=10{\cos }^{4}x=10{\left({\cos }^{2}x\right)}^{2}\hfill & \hfill \\ \text{ }=10{\left[\frac{1+\cos \left(2x\right)}{2}\right]}^{2}\hfill & {\text{Substitute reduction formula for cos}}^{2}x.\hfill \\ \text{ }=\frac{10}{4}\left[1+2\cos \left(2x\right)+{\cos }^{2}\left(2x\right)\right]\hfill & \hfill \\ \text{ }=\frac{10}{4}+\frac{10}{2}\cos \left(2x\right)+\frac{10}{4}\left(\frac{1+\cos 2\left(2x\right)}{2}\right)\hfill & {\text{Substitute reduction formula for cos}}^{2}x.\hfill \\ \text{ }=\frac{10}{4}+\frac{10}{2}\cos \left(2x\right)+\frac{10}{8}+\frac{10}{8}\cos \left(4x\right)\hfill & \hfill \\ \text{ }=\frac{30}{8}+5\cos \left(2x\right)+\frac{10}{8}\cos \left(4x\right)\hfill & \hfill \\ \text{ }=\frac{15}{4}+5\cos \left(2x\right)+\frac{5}{4}\cos \left(4x\right)\hfill & \hfill \end{array}[/latex]

5. [latex]-\frac{2}{\sqrt{5}}[/latex]

Solution to Odd-Numbered Exercises

1. Use the Pythagorean identities and isolate the squared term.

3. [latex]\frac{1-\cos x}{\sin x},\frac{\sin x}{1+\cos x}[/latex], multiplying the top and bottom by [latex]\sqrt{1-\cos x}[/latex] and [latex]\sqrt{1+\cos x}[/latex], respectively.

5. a) [latex]\frac{3\sqrt{7}}{32}[/latex] b) [latex]\frac{31}{32}[/latex] c) [latex]\frac{3\sqrt{7}}{31}[/latex]

7. a) [latex]\frac{\sqrt{3}}{2}[/latex] b) [latex]-\frac{1}{2}[/latex] c) [latex]-\sqrt{3}[/latex]

9. [latex]\cos \theta =-\frac{2\sqrt{5}}{5},\sin \theta =\frac{\sqrt{5}}{5},\tan \theta =-\frac{1}{2},\csc \theta =\sqrt{5},\sec \theta =-\frac{\sqrt{5}}{2},\cot \theta =-2[/latex]

11. [latex]2\sin \left(\frac{\pi }{2}\right)[/latex]

13. [latex]\frac{\sqrt{2-\sqrt{2}}}{2}[/latex]

15. [latex]\frac{\sqrt{2-\sqrt{3}}}{2}[/latex]

17. [latex]2+\sqrt{3}[/latex]

19. [latex]-1-\sqrt{2}[/latex]

21. a) [latex]\frac{3\sqrt{13}}{13}[/latex] b) [latex]-\frac{2\sqrt{13}}{13}[/latex] c) [latex]-\frac{3}{2}[/latex]

23. a) [latex]\frac{\sqrt{10}}{4}[/latex] b) [latex]\frac{\sqrt{6}}{4}[/latex] c) [latex]\frac{\sqrt{15}}{3}[/latex]

25. [latex]\frac{120}{169},-\frac{119}{169},-\frac{120}{119}[/latex]

27. [latex]\frac{2\sqrt{13}}{13},\frac{3\sqrt{13}}{13},\frac{2}{3}[/latex]

29. [latex]\cos \left({74}^{\circ }\right)[/latex]

31. [latex]\cos \left(18x\right)[/latex]

33. [latex]3\sin \left(10x\right)[/latex]

35. [latex]-2\sin \left(-x\right)\cos \left(-x\right)=-2\left(-\sin \left(x\right)\cos \left(x\right)\right)=\sin \left(2x\right)[/latex]

37. [latex]\begin{array}{l}\frac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)}{\tan }^{2}\theta =\frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{1+{\cos }^{2}\theta -{\sin }^{2}\theta }{\tan }^{2}\theta =\\ \frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{2{\cos }^{2}\theta }{\tan }^{2}\theta =\frac{\sin \left(\theta \right)}{\cos \theta }{\tan }^{2}\theta =\\ \cot \left(\theta \right){\tan }^{2}\theta =\tan \theta \end{array}[/latex]

39. [latex]\frac{1+\cos \left(12x\right)}{2}[/latex]

41. [latex]\frac{3+\cos \left(12x\right)-4\cos \left(6x\right)}{8}[/latex]

43. [latex]\frac{2+\cos \left(2x\right)-2\cos \left(4x\right)-\cos \left(6x\right)}{32}[/latex]

45. [latex]\frac{3+\cos \left(4x\right)-4\cos \left(2x\right)}{3+\cos \left(4x\right)+4\cos \left(2x\right)}[/latex]

47. [latex]\frac{1-\cos \left(4x\right)}{8}[/latex]

49. [latex]\frac{3+\cos \left(4x\right)-4\cos \left(2x\right)}{4\left(\cos \left(2x\right)+1\right)}[/latex]

51. [latex]\frac{\left(1+\cos \left(4x\right)\right)\sin x}{2}[/latex]

53. [latex]4\sin x\cos x\left({\cos }^{2}x-{\sin }^{2}x\right)[/latex]

55. [latex]\frac{2\tan x}{1+{\tan }^{2}x}=\frac{\frac{2\sin x}{\cos x}}{1+\frac{{\sin }^{2}x}{{\cos }^{2}x}}=\frac{\frac{2\sin x}{\cos x}}{\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}}=[/latex]

[latex]\frac{2\sin x}{\cos x}.\frac{{\cos }^{2}x}{1}=2\sin x\cos x=\sin \left(2x\right)[/latex]

57. [latex]\frac{2\sin x\cos x}{2{\cos }^{2}x - 1}=\frac{\sin \left(2x\right)}{\cos \left(2x\right)}=\tan \left(2x\right)[/latex]

59. [latex]\begin{array}{l}\sin \left(x+2x\right)=\sin x\cos \left(2x\right)+\sin \left(2x\right)\cos x\hfill \\ =\sin x\left({\cos }^{2}x-{\sin }^{2}x\right)+2\sin x\cos x\cos x\hfill \\ =\sin x{\cos }^{2}x-{\sin }^{3}x+2\sin x{\cos }^{2}x\hfill \\ =3\sin x{\cos }^{2}x-{\sin }^{3}x\hfill \end{array}[/latex]

61. [latex]\begin{array}{l}\frac{1+\cos \left(2t\right)}{\sin \left(2t\right)-\cos t}=\frac{1+2{\cos }^{2}t - 1}{2\sin t\cos t-\cos t}\hfill \\ =\frac{2{\cos }^{2}t}{\cos t\left(2\sin t - 1\right)}\hfill \\ =\frac{2\cos t}{2\sin t - 1}\hfill \end{array}[/latex]

63. [latex]\begin{array}{l}\left({\cos }^{2}\left(4x\right)-{\sin }^{2}\left(4x\right)-\sin \left(8x\right)\right)\left({\cos }^{2}\left(4x\right)-{\sin }^{2}\left(4x\right)+\sin \left(8x\right)\right)=\hfill \\ \text{ }=\left(\cos \left(8x\right)-\sin \left(8x\right)\right)\left(\cos \left(8x\right)+\sin \left(8x\right)\right)\hfill \\ \text{ }={\cos }^{2}\left(8x\right)-{\sin }^{2}\left(8x\right)\hfill \\ \text{ }=\cos \left(16x\right)\hfill \\ \hfill \end{array}[/latex]