{"id":1397,"date":"2023-06-05T14:51:19","date_gmt":"2023-06-05T14:51:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-double-angle-half-angle-and-reduction-formulas\/"},"modified":"2023-06-05T14:51:19","modified_gmt":"2023-06-05T14:51:19","slug":"solutions-for-double-angle-half-angle-and-reduction-formulas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-double-angle-half-angle-and-reduction-formulas\/","title":{"raw":"Solutions 53: Double-Angle, Half-Angle, and Reduction Formulas","rendered":"Solutions 53: Double-Angle, Half-Angle, and Reduction Formulas"},"content":{"raw":"\n

Solution to Odd-Numbered Exercises<\/h2>\n1. Use the Pythagorean identities and isolate the squared term.\n\n3. [latex]\\frac{1-\\cos x}{\\sin x},\\frac{\\sin x}{1+\\cos x}[\/latex], multiplying the top and bottom by [latex]\\sqrt{1-\\cos x}[\/latex] and [latex]\\sqrt{1+\\cos x}[\/latex], respectively.\n\n5. a) [latex]\\frac{3\\sqrt{7}}{32}[\/latex] b) [latex]\\frac{31}{32}[\/latex] c) [latex]\\frac{3\\sqrt{7}}{31}[\/latex]\n\n7. a) [latex]\\frac{\\sqrt{3}}{2}[\/latex] b) [latex]-\\frac{1}{2}[\/latex] c) [latex]-\\sqrt{3}[\/latex]\n\n9. [latex]\\cos \\theta =-\\frac{2\\sqrt{5}}{5},\\sin \\theta =\\frac{\\sqrt{5}}{5},\\tan \\theta =-\\frac{1}{2},\\csc \\theta =\\sqrt{5},\\sec \\theta =-\\frac{\\sqrt{5}}{2},\\cot \\theta =-2[\/latex]\n\n11. [latex]2\\sin \\left(\\frac{\\pi }{2}\\right)[\/latex]\n\n13. [latex]\\frac{\\sqrt{2-\\sqrt{2}}}{2}[\/latex]\n\n15. [latex]\\frac{\\sqrt{2-\\sqrt{3}}}{2}[\/latex]\n\n17. [latex]2+\\sqrt{3}[\/latex]\n\n19. [latex]-1-\\sqrt{2}[\/latex]\n\n21. a) [latex]\\frac{3\\sqrt{13}}{13}[\/latex] b) [latex]-\\frac{2\\sqrt{13}}{13}[\/latex] c) [latex]-\\frac{3}{2}[\/latex]\n\n23. a) [latex]\\frac{\\sqrt{10}}{4}[\/latex] b) [latex]\\frac{\\sqrt{6}}{4}[\/latex] c) [latex]\\frac{\\sqrt{15}}{3}[\/latex]\n\n25. [latex]\\frac{120}{169},-\\frac{119}{169},-\\frac{120}{119}[\/latex]\n\n27. [latex]\\frac{2\\sqrt{13}}{13},\\frac{3\\sqrt{13}}{13},\\frac{2}{3}[\/latex]\n\n29. [latex]\\cos \\left({74}^{\\circ }\\right)[\/latex]\n\n31. [latex]\\cos \\left(18x\\right)[\/latex]\n\n33. [latex]3\\sin \\left(10x\\right)[\/latex]\n\n35. [latex]-2\\sin \\left(-x\\right)\\cos \\left(-x\\right)=-2\\left(-\\sin \\left(x\\right)\\cos \\left(x\\right)\\right)=\\sin \\left(2x\\right)[\/latex]\n\n37. [latex]\\begin{array}{l}\\frac{\\sin \\left(2\\theta \\right)}{1+\\cos \\left(2\\theta \\right)}{\\tan }^{2}\\theta =\\frac{2\\sin \\left(\\theta \\right)\\cos \\left(\\theta \\right)}{1+{\\cos }^{2}\\theta -{\\sin }^{2}\\theta }{\\tan }^{2}\\theta =\\\\ \\frac{2\\sin \\left(\\theta \\right)\\cos \\left(\\theta \\right)}{2{\\cos }^{2}\\theta }{\\tan }^{2}\\theta =\\frac{\\sin \\left(\\theta \\right)}{\\cos \\theta }{\\tan }^{2}\\theta =\\\\ \\cot \\left(\\theta \\right){\\tan }^{2}\\theta =\\tan \\theta \\end{array}[\/latex]\n\n39. [latex]\\frac{1+\\cos \\left(12x\\right)}{2}[\/latex]\n\n41. [latex]\\frac{3+\\cos \\left(12x\\right)-4\\cos \\left(6x\\right)}{8}[\/latex]\n\n43. [latex]\\frac{2+\\cos \\left(2x\\right)-2\\cos \\left(4x\\right)-\\cos \\left(6x\\right)}{32}[\/latex]\n\n45. [latex]\\frac{3+\\cos \\left(4x\\right)-4\\cos \\left(2x\\right)}{3+\\cos \\left(4x\\right)+4\\cos \\left(2x\\right)}[\/latex]\n\n47. [latex]\\frac{1-\\cos \\left(4x\\right)}{8}[\/latex]\n\n49. [latex]\\frac{3+\\cos \\left(4x\\right)-4\\cos \\left(2x\\right)}{4\\left(\\cos \\left(2x\\right)+1\\right)}[\/latex]\n\n51. [latex]\\frac{\\left(1+\\cos \\left(4x\\right)\\right)\\sin x}{2}[\/latex]\n\n53. [latex]4\\sin x\\cos x\\left({\\cos }^{2}x-{\\sin }^{2}x\\right)[\/latex]\n\n55. [latex]\\frac{2\\tan x}{1+{\\tan }^{2}x}=\\frac{\\frac{2\\sin x}{\\cos x}}{1+\\frac{{\\sin }^{2}x}{{\\cos }^{2}x}}=\\frac{\\frac{2\\sin x}{\\cos x}}{\\frac{{\\cos }^{2}x+{\\sin }^{2}x}{{\\cos }^{2}x}}=[\/latex]\n[latex]\\frac{2\\sin x}{\\cos x}.\\frac{{\\cos }^{2}x}{1}=2\\sin x\\cos x=\\sin \\left(2x\\right)[\/latex]\n\n57. [latex]\\frac{2\\sin x\\cos x}{2{\\cos }^{2}x - 1}=\\frac{\\sin \\left(2x\\right)}{\\cos \\left(2x\\right)}=\\tan \\left(2x\\right)[\/latex]\n\n59. [latex]\\begin{array}{l}\\sin \\left(x+2x\\right)=\\sin x\\cos \\left(2x\\right)+\\sin \\left(2x\\right)\\cos x\\hfill \\\\ =\\sin x\\left({\\cos }^{2}x-{\\sin }^{2}x\\right)+2\\sin x\\cos x\\cos x\\hfill \\\\ =\\sin x{\\cos }^{2}x-{\\sin }^{3}x+2\\sin x{\\cos }^{2}x\\hfill \\\\ =3\\sin x{\\cos }^{2}x-{\\sin }^{3}x\\hfill \\end{array}[\/latex]\n\n61. [latex]\\begin{array}{l}\\frac{1+\\cos \\left(2t\\right)}{\\sin \\left(2t\\right)-\\cos t}=\\frac{1+2{\\cos }^{2}t - 1}{2\\sin t\\cos t-\\cos t}\\hfill \\\\ =\\frac{2{\\cos }^{2}t}{\\cos t\\left(2\\sin t - 1\\right)}\\hfill \\\\ =\\frac{2\\cos t}{2\\sin t - 1}\\hfill \\end{array}[\/latex]\n\n63. [latex]\\begin{array}{l}\\left({\\cos }^{2}\\left(4x\\right)-{\\sin }^{2}\\left(4x\\right)-\\sin \\left(8x\\right)\\right)\\left({\\cos }^{2}\\left(4x\\right)-{\\sin }^{2}\\left(4x\\right)+\\sin \\left(8x\\right)\\right)=\\hfill \\\\ \\text{ }=\\left(\\cos \\left(8x\\right)-\\sin \\left(8x\\right)\\right)\\left(\\cos \\left(8x\\right)+\\sin \\left(8x\\right)\\right)\\hfill \\\\ \\text{ }={\\cos }^{2}\\left(8x\\right)-{\\sin }^{2}\\left(8x\\right)\\hfill \\\\ \\text{ }=\\cos \\left(16x\\right)\\hfill \\\\ \\hfill \\end{array}[\/latex]\n","rendered":"

Solution to Odd-Numbered Exercises<\/h2>\n

1. Use the Pythagorean identities and isolate the squared term.<\/p>\n

3. [latex]\\frac{1-\\cos x}{\\sin x},\\frac{\\sin x}{1+\\cos x}[\/latex], multiplying the top and bottom by [latex]\\sqrt{1-\\cos x}[\/latex] and [latex]\\sqrt{1+\\cos x}[\/latex], respectively.<\/p>\n

5. a) [latex]\\frac{3\\sqrt{7}}{32}[\/latex] b) [latex]\\frac{31}{32}[\/latex] c) [latex]\\frac{3\\sqrt{7}}{31}[\/latex]<\/p>\n

7. a) [latex]\\frac{\\sqrt{3}}{2}[\/latex] b) [latex]-\\frac{1}{2}[\/latex] c) [latex]-\\sqrt{3}[\/latex]<\/p>\n

9. [latex]\\cos \\theta =-\\frac{2\\sqrt{5}}{5},\\sin \\theta =\\frac{\\sqrt{5}}{5},\\tan \\theta =-\\frac{1}{2},\\csc \\theta =\\sqrt{5},\\sec \\theta =-\\frac{\\sqrt{5}}{2},\\cot \\theta =-2[\/latex]<\/p>\n

11. [latex]2\\sin \\left(\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n

13. [latex]\\frac{\\sqrt{2-\\sqrt{2}}}{2}[\/latex]<\/p>\n

15. [latex]\\frac{\\sqrt{2-\\sqrt{3}}}{2}[\/latex]<\/p>\n

17. [latex]2+\\sqrt{3}[\/latex]<\/p>\n

19. [latex]-1-\\sqrt{2}[\/latex]<\/p>\n

21. a) [latex]\\frac{3\\sqrt{13}}{13}[\/latex] b) [latex]-\\frac{2\\sqrt{13}}{13}[\/latex] c) [latex]-\\frac{3}{2}[\/latex]<\/p>\n

23. a) [latex]\\frac{\\sqrt{10}}{4}[\/latex] b) [latex]\\frac{\\sqrt{6}}{4}[\/latex] c) [latex]\\frac{\\sqrt{15}}{3}[\/latex]<\/p>\n

25. [latex]\\frac{120}{169},-\\frac{119}{169},-\\frac{120}{119}[\/latex]<\/p>\n

27. [latex]\\frac{2\\sqrt{13}}{13},\\frac{3\\sqrt{13}}{13},\\frac{2}{3}[\/latex]<\/p>\n

29. [latex]\\cos \\left({74}^{\\circ }\\right)[\/latex]<\/p>\n

31. [latex]\\cos \\left(18x\\right)[\/latex]<\/p>\n

33. [latex]3\\sin \\left(10x\\right)[\/latex]<\/p>\n

35. [latex]-2\\sin \\left(-x\\right)\\cos \\left(-x\\right)=-2\\left(-\\sin \\left(x\\right)\\cos \\left(x\\right)\\right)=\\sin \\left(2x\\right)[\/latex]<\/p>\n

37. [latex]\\begin{array}{l}\\frac{\\sin \\left(2\\theta \\right)}{1+\\cos \\left(2\\theta \\right)}{\\tan }^{2}\\theta =\\frac{2\\sin \\left(\\theta \\right)\\cos \\left(\\theta \\right)}{1+{\\cos }^{2}\\theta -{\\sin }^{2}\\theta }{\\tan }^{2}\\theta =\\\\ \\frac{2\\sin \\left(\\theta \\right)\\cos \\left(\\theta \\right)}{2{\\cos }^{2}\\theta }{\\tan }^{2}\\theta =\\frac{\\sin \\left(\\theta \\right)}{\\cos \\theta }{\\tan }^{2}\\theta =\\\\ \\cot \\left(\\theta \\right){\\tan }^{2}\\theta =\\tan \\theta \\end{array}[\/latex]<\/p>\n

39. [latex]\\frac{1+\\cos \\left(12x\\right)}{2}[\/latex]<\/p>\n

41. [latex]\\frac{3+\\cos \\left(12x\\right)-4\\cos \\left(6x\\right)}{8}[\/latex]<\/p>\n

43. [latex]\\frac{2+\\cos \\left(2x\\right)-2\\cos \\left(4x\\right)-\\cos \\left(6x\\right)}{32}[\/latex]<\/p>\n

45. [latex]\\frac{3+\\cos \\left(4x\\right)-4\\cos \\left(2x\\right)}{3+\\cos \\left(4x\\right)+4\\cos \\left(2x\\right)}[\/latex]<\/p>\n

47. [latex]\\frac{1-\\cos \\left(4x\\right)}{8}[\/latex]<\/p>\n

49. [latex]\\frac{3+\\cos \\left(4x\\right)-4\\cos \\left(2x\\right)}{4\\left(\\cos \\left(2x\\right)+1\\right)}[\/latex]<\/p>\n

51. [latex]\\frac{\\left(1+\\cos \\left(4x\\right)\\right)\\sin x}{2}[\/latex]<\/p>\n

53. [latex]4\\sin x\\cos x\\left({\\cos }^{2}x-{\\sin }^{2}x\\right)[\/latex]<\/p>\n

55. [latex]\\frac{2\\tan x}{1+{\\tan }^{2}x}=\\frac{\\frac{2\\sin x}{\\cos x}}{1+\\frac{{\\sin }^{2}x}{{\\cos }^{2}x}}=\\frac{\\frac{2\\sin x}{\\cos x}}{\\frac{{\\cos }^{2}x+{\\sin }^{2}x}{{\\cos }^{2}x}}=[\/latex]
\n[latex]\\frac{2\\sin x}{\\cos x}.\\frac{{\\cos }^{2}x}{1}=2\\sin x\\cos x=\\sin \\left(2x\\right)[\/latex]<\/p>\n

57. [latex]\\frac{2\\sin x\\cos x}{2{\\cos }^{2}x - 1}=\\frac{\\sin \\left(2x\\right)}{\\cos \\left(2x\\right)}=\\tan \\left(2x\\right)[\/latex]<\/p>\n

59. [latex]\\begin{array}{l}\\sin \\left(x+2x\\right)=\\sin x\\cos \\left(2x\\right)+\\sin \\left(2x\\right)\\cos x\\hfill \\\\ =\\sin x\\left({\\cos }^{2}x-{\\sin }^{2}x\\right)+2\\sin x\\cos x\\cos x\\hfill \\\\ =\\sin x{\\cos }^{2}x-{\\sin }^{3}x+2\\sin x{\\cos }^{2}x\\hfill \\\\ =3\\sin x{\\cos }^{2}x-{\\sin }^{3}x\\hfill \\end{array}[\/latex]<\/p>\n

61. [latex]\\begin{array}{l}\\frac{1+\\cos \\left(2t\\right)}{\\sin \\left(2t\\right)-\\cos t}=\\frac{1+2{\\cos }^{2}t - 1}{2\\sin t\\cos t-\\cos t}\\hfill \\\\ =\\frac{2{\\cos }^{2}t}{\\cos t\\left(2\\sin t - 1\\right)}\\hfill \\\\ =\\frac{2\\cos t}{2\\sin t - 1}\\hfill \\end{array}[\/latex]<\/p>\n

63. [latex]\\begin{array}{l}\\left({\\cos }^{2}\\left(4x\\right)-{\\sin }^{2}\\left(4x\\right)-\\sin \\left(8x\\right)\\right)\\left({\\cos }^{2}\\left(4x\\right)-{\\sin }^{2}\\left(4x\\right)+\\sin \\left(8x\\right)\\right)=\\hfill \\\\ \\text{ }=\\left(\\cos \\left(8x\\right)-\\sin \\left(8x\\right)\\right)\\left(\\cos \\left(8x\\right)+\\sin \\left(8x\\right)\\right)\\hfill \\\\ \\text{ }={\\cos }^{2}\\left(8x\\right)-{\\sin }^{2}\\left(8x\\right)\\hfill \\\\ \\text{ }=\\cos \\left(16x\\right)\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/p>\n\n\t\t\t

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