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
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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. \(\tan(6t)\)