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Mathematics LibreTexts

7.2E: Exercises - Sum and Difference Identities

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    74425
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    A: Evaluate sum and difference formulas from a given angle

    Exercise \(\PageIndex{A}\)

    \( \bigstar \) Find the exact value.

    1. \( \cos \left ( \dfrac{7\pi }{4}- \dfrac{5\pi }{3}  \right) \\[4pt]\)

    2. \( \cos \left ( \dfrac{11\pi }{6}- \dfrac{3\pi }{4}  \right) \\[4pt]\)

    3. \( \cos \left ( \dfrac{\pi }{4}+ \dfrac{2\pi }{3}  \right) \\[4pt]\)

    4. \( \cos \left ( \dfrac{5\pi }{4}+ \dfrac{\pi }{3}  \right) \\[4pt]\)

    5. \( \sin \left ( \dfrac{5\pi }{4}- \dfrac{7\pi }{6}  \right) \\[4pt]\)

    6. \( \sin \left ( \dfrac{5\pi }{3}- \dfrac{3\pi }{4}  \right) \\[4pt]\)

    7. \( \sin \left ( \dfrac{\pi }{4}+ \dfrac{5\pi }{6}  \right) \\[4pt]\)

    8. \( \sin \left ( \dfrac{7\pi }{4}+ \dfrac{\pi }{6}  \right) \\[4pt]\)

    9. \( \tan \left ( \dfrac{11\pi }{6}- \dfrac{5\pi }{4}  \right) \\[4pt]\)

    10. \( \tan \left ( \dfrac{7\pi }{4}- \dfrac{2\pi }{3}  \right) \\[4pt]\)

    11. \( \tan \left ( \dfrac{\pi }{4}+ \dfrac{\pi }{6}  \right) \\[4pt]\)

    12. \( \tan \left ( \dfrac{3\pi }{4}+ \dfrac{7\pi }{6}  \right) \\[4pt]\)

    \( \bigstar \) Find the exact value. 

    13. \( \cos \left (\dfrac{\pi }{12} \right) = \cos \left ( \dfrac{4\pi }{3}- \dfrac{5\pi }{4}  \right) \\[4pt] \)

    14. \(\cos \left (\dfrac{7\pi }{12} \right) = \cos \left ( \dfrac{5\pi }{6}- \dfrac{\pi }{4}  \right) \\[4pt] \)

    15. \( \sin \left (\dfrac{11\pi }{12} \right) = \sin \left ( \dfrac{3\pi }{4}+ \dfrac{\pi }{6}  \right)  \)

    16. \(\sin \left (\dfrac{5\pi }{12} \right) = \sin \left ( \dfrac{7\pi }{4}- \dfrac{4\pi }{3}  \right)  \\[4pt]  \)

    17. \(\tan \left (\dfrac{19\pi }{12} \right) =  \tan \left ( \dfrac{3\pi }{4}+ \dfrac{5\pi }{6}  \right) \\[4pt]\)

    18. \(\tan \left (-\dfrac{\pi }{12} \right) =  \tan \left ( \dfrac{7\pi }{6}- \dfrac{5\pi }{4}  \right) \)

    19. \(\sin (195^{\circ})  \)

    20. \(\sin (75^{\circ})  \)

    21. \(\cos (345^{\circ})  \)

    22. \(\cos (165^{\circ})   \)

    23. \(\tan (-15^{\circ}) \)

    24. \(\cos (105^{\circ}) \)

    \( \bigstar \) Simplify

    25. \(\sin \left (x-\dfrac{3\pi }{4} \right)\) 26. \(\sin \left (x+\dfrac{11\pi }{6} \right)\) 27. \(\cos \left (x+\dfrac{2\pi }{3} \right)\) 28. \(\cos \left (x-\dfrac{5\pi }{6} \right)\)
    29. \(\sec \left (\dfrac{\pi }{2}-\theta \right)\) 30. \(\csc \left (\dfrac{\pi }{2}-t \right)\) 31. \(\cot \left (\dfrac{\pi }{2}-x \right)\) 32. \(\tan \left (\dfrac{\pi }{4}-x \right)\)

    \( \bigstar \) (a) Simplify and then (b) graph

    33. \(\cos \left ( \dfrac{\pi }{2}-x \right )\)

    34. \(\sin (\pi -x)\)

    35. \(\tan \left ( \dfrac{\pi }{3}+x \right )\)

    36. \(\sin \left ( \dfrac{\pi }{3}+x \right )\)

    37. \(\tan \left ( \dfrac{\pi }{4}-x \right )\)

    38. \(\cos \left ( \dfrac{7\pi }{6}+x \right )\)

    39. \(\sin \left ( \dfrac{\pi }{4}+x \right )\)

    40. \(\cos \left ( \dfrac{5\pi }{4}+x \right )\)

    Answers to odd exercises.

    1.  \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)     3.  \(\dfrac{-\sqrt{2}-\sqrt{6}}{4}\)    5.  \(\dfrac{\sqrt{6}-\sqrt{2}}{4}\)    7.  \(\dfrac{\sqrt{2}-\sqrt{6}}{4}\)        9. \(  -2 - \sqrt{3}  \)     11. \(  2 + \sqrt{3}  \)

    13. \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)     15. \(\dfrac{\sqrt{6}-\sqrt{2}}{4}\)     17. \(-2-\sqrt{3}\)  19. \(-\dfrac{\sqrt{3}-1}{2\sqrt{2}}\)     21. \(\dfrac{1+\sqrt{3}}{2\sqrt{2}}\)      23. \(  -2 + \sqrt{3}  \)

    25. \(-\dfrac{\sqrt{2}}{2}\sin x-\dfrac{\sqrt{2}}{2}\cos x\)      27. \(-\dfrac{1}{2}\cos x-\dfrac{\sqrt{3}}{2}\sin x\)     29. \(\csc \theta\)     31. \(\tan x\)

    33. \(\sin x\)

    CNX_Precalc_Figure_07_02_201.jpg

    35. \(\cot \left ( \dfrac{\pi }{6}-x \right )\)

    CNX_Precalc_Figure_07_02_203.jpg

    37. \(\cot \left ( \dfrac{\pi }{4}+x \right )\)

    CNX_Precalc_Figure_07_02_205.jpg

    39. \(\dfrac{\sin x}{\sqrt{2}}+\dfrac{\cos x}{\sqrt{2}}\)

    CNX_Precalc_Figure_07_02_207.jpg

    B: Evaluate sum and difference formulas given trig ratios of angles

    Exercise \(\PageIndex{B}\)

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

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

    42. Given that \(\sin a=\dfrac{2}{3}\) and \(\cos b=-\dfrac{1}{4}\), with \(a\) and \(b\) both in the interval \(\left [ \dfrac{\pi }{2}, \pi \right )\),

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

    43. Angles \(A\) and \(B\) are in standard position and \(\sin( A ) = \dfrac{1}{2}, \cos(A) > 0)\), \(\cos(B) = \dfrac{3}{4}\), and \(\sin(B) < 0\).
          Draw a picture of the angles \(A\) and \(B\) in the plane and then find each of the following.

    (a) \(\cos(A + B)\)          (b) \(\cos(A - B)\)          (c) \(\sin(A + B)\)          (d) \(\sin(A - B)\)          (e) \(\tan(A + B)\)          (f) \(\tan(A - B)\)  

    \( \bigstar \) Given the following information about angles \(A\) and \(B\), find the exact value for each of the following.

     (a) \(\cos(A + B)\)          (b) \(\cos(A - B)\)          (c) \(\sin(A + B)\)          (d) \(\sin(A - B)\)          (e) \(\tan(A + B)\)          (f) \(\tan(A - B)\)  

    45. \( \sin A = -\dfrac{3}{5} \) with \(A\) in Quadrant III, and \( \cos B = \dfrac{1}{2} \) with  \(B\) in Quadrant IV.

    46. \( \sin A =  \dfrac{4}{5} \) with \(A\) in Quadrant I, and \( \tan B = -\dfrac{\sqrt{5}}{2} \) with  \(B\) in Quadrant II.

    47. \( \cos A = \dfrac{1}{2} \) with \( 0 \le A \le \tfrac{\pi}{2}  \), and \( \tan B  = 2 \sqrt{2} \) with  \( \pi \le B \le \tfrac{3\pi}{2} \).

    48. \( \cos A = -\dfrac{\sqrt{3}}{3} \) with \( \pi \le A \le \tfrac{3\pi}{2} \), and \( \sin B = \dfrac{\sqrt{3}}{3} \) with  \(\tfrac{\pi}{2} \le B \le \pi \).

    49. \( A = \tan^{-1} \left(  \sqrt{5} \right) \) and  \( B = \sin^{-1} \left( - \dfrac{\sqrt{5}}{3} \right) \)

    50. \( A = \tan^{-1} \left( - 2 \right) \) and \( B = \cos^{-1} \left( - \dfrac{\sqrt{6}}{6} \right) \) 

    \( \bigstar \) Find the exact value of each expression.

    52.
    \( \quad \sin \left ( \cos^{-1}\left ( 0 \right )- \cos^{-1}\left ( \dfrac{1}{2} \right )\right )\)

    53.
    \( \quad \cos \left ( \cos^{-1}\left ( \dfrac{\sqrt{2}}{2} \right )+ \sin^{-1}\left ( \dfrac{\sqrt{3}}{2} \right )\right )\)

    54.
    \( \quad \tan \left ( \sin^{-1}\left ( \dfrac{1}{2} \right )- \cos^{-1}\left ( \dfrac{1}{2} \right )\right )\)

     

    58. \( \tan(\sin^{-1} u + \cos^{-1} u ) \)

    59. \( \cos( \sin^{-1} u + \tan^{-1} u ) \)

    60. \( \sin( \cos^{-1} u +\ tan^{-1} u )  \)

    Answers to odd exercises.

    41a. \(\sin (a-b)=\left ( \dfrac{4}{5} \right )\left ( \dfrac{1}{3} \right )-\left ( \dfrac{3}{5} \right )\left ( \dfrac{2\sqrt{2}}{3} \right )=\dfrac{4-6\sqrt{2}}{15}\)

    41b. \(\cos (a+b)=\left ( \dfrac{3}{5} \right )\left ( \dfrac{1}{3} \right )-\left ( \dfrac{4}{5} \right )\left ( \dfrac{2\sqrt{2}}{3} \right )=\dfrac{3-8\sqrt{2}}{15}\)

    43. Use the Pythagorean Identity to determine  \(\cos(A) = \dfrac{\sqrt{3}}{2}\) and \(\sin(B) = -\dfrac{\sqrt{7}}{4}\)
          43(a). \(\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) = \dfrac{\sqrt{3}}{2}\cdot\dfrac{3}{4} - \dfrac{1}{2}\cdot(-\dfrac{\sqrt{7}}{4}) = \dfrac{3\sqrt{3} + \sqrt{7}}{8}\)

    45. \( \sin (A+B) = \tfrac{-3}{5} \cdot \tfrac{-4}{5} + \tfrac{-4}{5} \cdot \tfrac{\sqrt{3}}{2} = \dfrac{-3-4\sqrt{3}}{10} \), \( \cos(A+B) = \dfrac{-4+3\sqrt{3}}{10} \), \( \tan(A+B) = \dfrac{48 + 25\sqrt{3}}{-11} \\ \)
    \( \quad \;\; \sin(A-B) = \dfrac{-3+4\sqrt{3}}{10} \), \( \cos(A-B) =  \dfrac{-4-3\sqrt{3}}{10} \), \( \tan(A-B) =  \dfrac{48 + 25\sqrt{3}}{-11} \)

    47. \( \sin (A+B) = \tfrac{\sqrt{3}}{2} \cdot \tfrac{-1}{3} + \tfrac{1}{2} \cdot \tfrac{-2\sqrt{2}}{3} = \dfrac{-\sqrt{3}-2\sqrt{2}}{6}\), \( \cos(A+B) = \dfrac{-1+2\sqrt{6}}{6}\), \( \tan(A+B) = \dfrac{-9\sqrt{3} - 8\sqrt{2}}{23} \\ \)
    \( \quad \;\; \sin(A-B) = \dfrac{-\sqrt{3}+2\sqrt{2}}{6} \), \( \cos(A-B) =  \dfrac{-1-2\sqrt{6}}{6}\), \( \tan(A-B) = \dfrac{-9\sqrt{3} + 8\sqrt{2}}{23} \)

    49. \( \sin (A+B) = \tfrac{\sqrt{3}}{6} \cdot \tfrac{2}{3} + \tfrac{\sqrt{6}}{6} \cdot \tfrac{-\sqrt{5}}{3} = \dfrac{\sqrt{30}}{18},\) \( \cos(A+B) = \dfrac{7\sqrt{6}}{18},\) \( \tan(A+B) = \dfrac{\sqrt{5}}{7} \\ \) 
    \( \quad \;\; \sin(A-B) = \dfrac{\sqrt{30}}{6}\), \( \cos(A-B) = \dfrac{-\sqrt{6}}{6}\), \( \tan(A-B) = -\sqrt{5} \)

    53. \(\dfrac{\sqrt{2}-\sqrt{6}}{4}\)       59. \(  \dfrac{\sqrt{1-u^4}}{1+u^2} -u^2 \)

    C: Solve Equations

    Exercise \(\PageIndex{D}\)

    \( \bigstar \) Solve each equation for all solutions.

    65. \(\sin \left(3x\right)\cos \left(6x\right)-\cos \left(3x\right)\sin \left(6x\right)= -0.9\)

    66. \(\sin \left(6x\right)\cos \left(11x\right)-\cos \left(6x\right)\sin \left(11x\right)= -0.1\)

    67. \(\cos \left(2x\right)\cos \left(x\right)+\sin \left(2x\right)\sin \left(x\right)=1\)

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

    Answers to odd exercises.

    65. \(0.373 + \dfrac{2\pi}{3} k\) and \(0.674 + \dfrac{2\pi}{3} k\), where \(k\) is an integer     67. \(2 \pi k\), where \(k\) is an integer

    D: Verify Identities

    Exercise \(\PageIndex{F}\)

    \( \bigstar \) Simplify.

    71.  \(\dfrac{\tan \left (\dfrac{3}{2}x \right)-\tan \left (\dfrac{7}{5}x \right)}{1+\tan \left (\dfrac{3}{2}x \right)\tan \left (\dfrac{7}{5}x \right)}\) 72. \(\sin(2x)\cos(5x)-\sin(5x)\cos(2x)\)

    \( \bigstar \) Verify the Identity.

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

    74. \(\dfrac{\sin\;(A-B)}{\sin\;(A+B)} ~=~ \dfrac{\cot\;B \;-\; \cot\;A}{\cot\;B \;+\; \cot\;A}\)

    75. \(\cot\;A ~+~ \cot\;B ~=~ \dfrac{\sin\;(A+B)}{\sin\;A~\sin\;B}\)

    76. \(\dfrac{\sin(r + s)}{\cos(r)\cos(s)} = \tan(r) + \tan(s)\)

    77. \(\dfrac{\sin(r - s)}{\cos(r)\cos(s)} = \tan(r) - \tan(s)\)

    78. \(\dfrac{\cos\;(A+B)}{\sin\;A~\cos\;B} ~=~ \cot\;A \;-\; \tan\;B\)

    79. \(\dfrac{\cos (a+b)}{\cos a \cos b}=1-\tan a \tan b\)

    80. \(\dfrac{\tan (a+b)}{\tan (a-b)}=\dfrac{\sin a \cos a + \sin b \cos b}{\sin a \cos a - \sin b \cos b}\)

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

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

    83. \(\dfrac{\tan (x+y)}{1+\tan x \tan x}=\dfrac{\tan x + \tan y}{1-\tan^2 x \tan^2 y}\)

    84. \(\cot\;(A+B) ~=~ \dfrac{\cot\;A~\cot\;B \;-\; 1}{\cot\;A \;+\; \cot\;B}\)

    85. \(\cot\;(A-B) ~=~ \dfrac{\cot\;A~\cot\;B \;+\; 1}{\cot\;B \;-\; \cot\;A}\)

    87. \(\dfrac{\cos(x+h)-\cos(x)}{h}=\cos x\dfrac{\cos (h)-1}{h}-\sin x \dfrac{\sin (h)}{h}\)

    Answers to SOME odd exercises
    71. \(\tan \left (\dfrac{x}{10} \right)\)

    75. \(\cot\;A ~+~ \cot\;B = \text{___________} = \dfrac{\sin\;(A+B)}{\sin\;A~\sin\;B}\)
    77. \(\dfrac{\sin(r - s)}{\cos(r)\cos(s)} = \text{___________} = \tan(r) - \tan(s)\)
    79. \( \dfrac{\cos (a+b)}{\cos a \cos b} =  \dfrac{\cos a \cos b}{\cos a \cos b}- \dfrac{\sin a \sin b}{\cos a \cos b}= 1-\tan a \tan b \)
    81. \( \tan \left ( x+\dfrac{\pi }{4} \right ) =  \dfrac{\tan x + \tan\left (\tfrac{\pi}{4} \right )}{1-\tan x \tan\left (\tfrac{\pi}{4} \right )} =  \dfrac{\tan x+1}{1-\tan x(1)} = \dfrac{\tan x+1}{1-\tan x} \)
    83. \(\dfrac{\tan (x+y)}{1+\tan x \tan x}= \text{___________} = \dfrac{\tan x + \tan y}{1-\tan^2 x \tan^2 y}\)
    85. \(\cot\;(A-B) = \text{___________} = \dfrac{\cot\;A~\cot\;B \;+\; 1}{\cot\;B \;-\; \cot\;A}\)
    87. \( \dfrac{\cos(x+h)-\cos(x)}{h} = \dfrac{\cos x\cos (h) - \sin x\sin (h) -\cos x}{h} = \dfrac{\cos x(\cos (h)-1) - \sin x(\sin (h))}{h} \\ \)
    \( \quad \;\; = \cos x \cdot \dfrac{\cos (h)-1}{h}-\sin x \cdot \dfrac{\sin (h)}{h} \)

    \( \bigstar \) Verify the Identity.

    90. \(2\sin \left(a+b\right)\sin \left(a-b\right)=\cos \left(2b\right)-{\rm cos}(2a)\)

    91. \(\cos(x+y)\cos(x-y)=\cos^2x-\sin^2y\)

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

    93. \( \sin(4x)-\sin(3x)\cos x =\sin x \cos(3x) \)

    94. \( \cos(4x)+\sin x \sin(3x) = \cos x \cos(3x)\)

    95. \( \sin(3x)\cos(6x)=\sin(9x)-\cos(3x)\sin(6x)\) 

    96. \( \sin(4x) = \sin(5x)\cos x-\cos(5x)\sin x  \)

    97. \(\sin(3x)=3\sin x \cos^2x-\sin^3x   \)

    98. \(\cos(3x)=\cos^3x-3\sin^2x\cos x    \)

    99. \( \sin(2x) = 2 \sin x \cos x\)

    100. \( \cos(2\theta ) = \cos^2\theta -\sin^2\theta\)

    101. \( \tan(2\theta ) = \dfrac{2\tan \theta }{1-\tan^2\theta }\)

    102. \( \tan(-x)=\dfrac{\tan x-\tan(2x)}{1+\tan x \tan(2x)} \)

    Answers to SOME odd exercises.
    91. \(\cos(x+y)\cos(x-y) = \text{___________} = \cos^2x-\sin^2y\)
    93. \( \sin(4x)-\sin(3x)\cos x = \text{___________} = \sin x \cos(3x) \)
    95. \( \sin(3x)\cos(6x) = \text{___________} = \sin(9x)-\cos(3x)\sin(6x)\)
    97. \( \sin (x+2x) = \sin x \cos (2x)+\sin (2x) \cos x = \sin x(\cos ^2 x - \sin ^2 x)+2\sin x \cos x \cos x \\ 
    = \sin x \cos ^2 x-\sin ^3 x + 2\sin x\cos ^2 x = 3\sin x\cos ^2 x - \sin ^3 x \)
    99. \( \sin(2x) = \text{___________} = 2 \sin x \cos x\)
    101. \( \tan(2\theta ) = \text{___________} = \dfrac{2\tan \theta }{1-\tan^2\theta }\)

    \( \bigstar \)

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