7.2E: Addition and Subtraction Identities (Exercises)

Section 7.2 Exercises

Find an exact value for each of the following.

1. $\sin \left(75{}^\circ \right)$

2. $\sin \left(195{}^\circ \right)$

3. ${\rm cos}(165{}^\circ )$

4. ${\rm cos}(345{}^\circ )$

5. $\cos \left(\dfrac{7\pi }{12} \right)$

6. $\cos \left(\dfrac{\pi }{12} \right)$

7. $\sin \left(\dfrac{5\pi }{12} \right)$

8. $\sin \left(\dfrac{11\pi }{12} \right)$

Rewrite in terms of $\sin \left(x\right)$ and $\cos \left(x\right)$.

9. $\sin \left(x+\dfrac{11\pi }{6} \right)$

10. $\sin \left(x-\dfrac{3\pi }{4} \right)$

11. $\cos \left(x-\dfrac{5\pi }{6} \right)$

12. $\cos \left(x+\dfrac{2\pi }{3} \right)$

Simplify each expression.

13. $\csc \left(\dfrac{\pi }{2} -\; t\right)$

14. $\sec \left(\dfrac{\pi }{2} -w\right)$

15. $\cot \left(\dfrac{\pi }{2} -x\right)$

16. $\tan \left(\dfrac{\pi }{2} -x\right)$

Rewrite the product as a sum.

17. $16\sin \left(16x\right)\sin \left(11x\right)$

18. $20\cos \left(36t\right)\cos \left(6t\right)$

19. $2\sin \left(5x\right)\cos \left(3x\right)$

20. $10\cos \left(5x\right)\sin \left(10x\right)$

Rewrite the sum as a product.

21. $\cos \left(6t\right)+\cos \left(4t\right)$

22. $\cos \left(6u\right)+\cos \left(4u\right)$

23. $\sin \left(3x\right)+\sin \left(7x\right)$

24. $\sin \left(h\right)+\sin \left(3h\right)$

25. Given $\sin \left(a\right)=\dfrac{2}{3}$ and $\cos \left(b\right)=-\dfrac{1}{4}$, with $a$ and $b$ both in the interval $\left[\dfrac{\pi }{2} ,\pi \right)$:

a. Find $\sin \left(a+b\right)$
b. Find $\cos \left(a-b\right)$

26. Given $\sin \left(a\right)=\dfrac{4}{5}$ and $\cos \left(b\right)=\dfrac{1}{3}$, with $a$ and $b$ both in the interval $\left[0,\dfrac{\pi }{2} \right)$:

a. Find $\sin \left(a-b\right)$
b. Find $\cos \left(a+b\right)$

Solve each equation for all solutions.

27. $\sin \left(3x\right)\cos \left(6x\right)-\cos \left(3x\right)\sin \left(6x\right)= -0.9$

28. $\sin \left(6x\right)\cos \left(11x\right)-\cos \left(6x\right)\sin \left(11x\right)= -0.1$

29. $\cos \left(2x\right)\cos \left(x\right)+\sin \left(2x\right)\sin \left(x\right)=1$

30. $\cos \left(5x\right)\cos \left(3x\right)-\sin \left(5x\right)\sin \left(3x\right)=\dfrac{\sqrt{3} }{2}$

Solve each equation for all solutions.

31. $\cos \left(5x\right)=-\cos \left(2x\right)$

32. $\sin \left(5x\right)=\sin \left(3x\right)$

33. $\cos \left(6\theta \right)-\cos \left(2\theta \right)=\sin \left(4\theta \right)$

34. $\cos \left(8\theta \right)-\cos \left(2\theta \right)=\sin \left(5\theta \right)$

Rewrite as a single function of the form $A\sin (Bx+C)$.

35. $4\sin \left(x\right)-6\cos \left(x\right)$

36. $-\sin \left(x\right)-5\cos \left(x\right)$

37. $5\sin \left(3x\right)+2\cos \left(3x\right)$

38. $-3\sin \left(5x\right)+4\cos \left(5x\right)$

Solve for the first two positive solutions.

39. $-5\sin \left(x\right)+3\cos \left(x\right)=1$

40. $3\sin \left(x\right)+\cos \left(x\right)=2$

41. $3\sin \left(2x\right)-5\cos \left(2x\right)=3$

42. $-3\sin \left(4x\right)-2\cos \left(4x\right)=1$

Simplify.

43. $\dfrac{\sin \left(7t\right)+\sin \left(5t\right)}{\cos \left(7t\right)+\cos \left(5t\right)}$

44. $\dfrac{\sin \left(9t\right)-\sin \left(3t\right)}{\cos \left(9t\right)+\cos \left(3t\right)}$

Prove the identity.

44. $\tan \left(x+\dfrac{\pi }{4} \right)=\dfrac{\tan \left(x\right)+1}{1-\tan \left(x\right)}$

45. $\tan \left(\dfrac{\pi }{4} -t\right)=\dfrac{1-\tan \left(t\right)}{1+\tan \left(t\right)}$

46. $\cos \left(a+b\right)+\cos \left(a-b\right)=2\cos \left(a\right)\cos \left(b\right)$

47. $\dfrac{\cos \left(a+b\right)}{\cos \left(a-b\right)} =\dfrac{1-\tan \left(a\right)\tan \left(b\right)}{1+\tan \left(a\right)\tan \left(b\right)}$

48. $\dfrac{\tan \left(a+b\right)}{\tan \left(a-b\right)} =\dfrac{\sin \left(a\right)\cos \left(a\right)+\sin \left(b\right)\cos \left(b\right)}{\sin \left(a\right)\cos \left(a\right)-\sin \left(b\right)\cos \left(b\right)}$

49. $2\sin \left(a+b\right)\sin \left(a-b\right)=\cos \left(2b\right)-{\rm cos}(2a)$

50. $\dfrac{\sin \left(x\right)+\sin \left(y\right)}{\cos \left(x\right)+\cos \left(y\right)} =\tan \left(\dfrac{1}{2} \left(x+y\right)\right)$

Prove the identity.

51. $\dfrac{\cos \left(a+b\right)}{\cos \left(a\right)\cos \left(b\right)} =1-\tan \left(a\right)\tan \left(b\right)$

52. $\cos \left(x+y\right)\cos \left(x-y\right)=\cos ^{2} x-\sin ^{2} y$

53. Use the sum and difference identities to establish the product-to-sum identity

$\sin (\alpha )\sin (\beta )=\dfrac{1}{2} \left(\cos (\alpha -\beta )-\cos (\alpha +\beta )\right)$

54. Use the sum and difference identities to establish the product-to-sum identity

$\cos (\alpha )\cos (\beta )=\dfrac{1}{2} \left(\cos (\alpha +\beta )+\cos (\alpha -\beta )\right)$

55. Use the product-to-sum identities to establish the sum-to-product identity

$\cos \left(u\right)+\cos \left(v\right)=2\cos \left(\dfrac{u+v}{2} \right)\cos \left(\dfrac{u-v}{2} \right)$

56. Use the product-to-sum identities to establish the sum-to-product identity

$\cos \left(u\right)-\cos \left(v\right)=-2\sin \left(\dfrac{u+v}{2} \right)\sin \left(\dfrac{u-v}{2} \right)$

Answer

1. $\dfrac{\sqrt{2} + \sqrt{6}}{4}$

3. $\dfrac{-\sqrt{2} - \sqrt{6}}{4}$

5. $\dfrac{\sqrt{2} - \sqrt{6}}{4}$

7. $\dfrac{\sqrt{2} + \sqrt{6}}{4}$

9. $\dfrac{\sqrt{3}}{2}\sin(x) - \dfrac{1}{2} \cos(x)$

11. $-\dfrac{\sqrt{3}}{2}\cos(x) + \dfrac{1}{2} \sin(x)$

13. $\sec(t)$

15. $\tan(x)$

17. $8(\cos(5x) - \cos(27x))$

19. $\sin(8x) + \sin (2x)$

21. $2 \cos(5t) \cos(t)$

23. $2 \sin(5x) \cos(2x)$

25. a. $(\dfrac{2}{3})(-\dfrac{1}{4}) + (-\dfrac{\sqrt{5}}{3})(\dfrac{\sqrt{15}}{4}) = \dfrac{-2-5\sqrt{3}}{12}$
b. $(-\dfrac{\sqrt{5}}{3})(-\dfrac{1}{4}) + (\dfrac{2}{3})(\dfrac{\sqrt{15}}{4}) = \dfrac{\sqrt{5} + 2\sqrt{15}}{12}$

27. $0.373 + \dfrac{2\pi}{3} k$ and $0.674 + \dfrac{2\pi}{3} k$, where $k$ is an integer

29. $2 \pi k$, where $k$ is an integer

31. $\dfrac{\pi}{7} + \dfrac{4\pi}{7} k$, $\dfrac{3\pi}{7} + \dfrac{4\pi}{7} k$, $\dfrac{\pi}{3} + \dfrac{4\pi}{3} k$, and $\pi + \dfrac{4\pi}{3} k$, where $k$ is an integer

33. $\dfrac{7\pi}{12} + \pi k$, $\dfrac{11\pi}{12} + \pi k$, and $\dfrac{\pi}{4} k$, where $k$ is an integer

35. $2 \sqrt{13} \sin (x + 5.3004)$ or $2\sqrt{13} \sin(x - 0.9828)$

37. $\sqrt{29} \sin(3x + 0.3805)$

39. 0.3681, 3.8544

41. 0.7854, 1.8158

43. $\tan(6t)$