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# 7.2.2E: Addition and Subtraction Identities (Exercises)

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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

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

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. $$\tan(6t)$$