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Mathematics LibreTexts

7.2E: Addition and Subtraction Identities (Exercises)

  • Page ID
    13935
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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. \(\cos \left(\frac{7\pi }{12} \right)\)

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

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

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

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

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

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

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

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

    Simplify each expression.

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

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

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

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

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

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

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

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

    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. \(\frac{\sin \left(7t\right)+\sin \left(5t\right)}{\cos \left(7t\right)+\cos \left(5t\right)}\)

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

    Prove the identity.

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

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

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

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

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

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

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

    Prove the identity.

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

    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

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

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

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

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

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

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

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

    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. \(\tan(6t)\)