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

    Section 7.3 Double Angle Identities 487