6.4e: Exercises - Sum and Difference Identities (2024)

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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.\(\tanx\)

33. \(\sin x\)

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

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

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

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 information about angles \(A\) and\(B\) in the exercises below, 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 QuadrantIII, and \( \cos B = \dfrac{1}{2} \) with\(B\)in Quadrant IV.

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

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

48. \( \cosA = -\dfrac{\sqrt{3}}{3} \) with\( \pi \le A \le \tfrac{3\pi}{2} \), and \( \sinB = \dfrac{\sqrt{3}}{3} \) 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. \(\sin (a-b)=\left ( \frac{4}{5} \right )\left ( \frac{1}{3} \right )-\left ( \frac{3}{5} \right )\left ( \frac{2\sqrt{2}}{3} \right )=\dfrac{4-6\sqrt{2}}{15}\)

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

43. a. \( \tfrac{\sqrt{3}}{2} \cdot \tfrac{3}{4} - \tfrac{1}{2} \cdot\tfrac{-\sqrt{7}}{4} =\dfrac{3\sqrt{3} + \sqrt{7}}{8}\), b. \( \dfrac{3\sqrt{3} - \sqrt{7}}{8}\), c.\( \dfrac{3- \sqrt{21}}{8}\), d.\( \dfrac{3+ \sqrt{21}}{8}\), e.\(\dfrac{4\sqrt{3} -3 \sqrt{7}}{5}\), f.\(\dfrac{4\sqrt{3} +3 \sqrt{7}}{5}\)

45. \( \sin (A+B) = \tfrac{-3}{5} \cdot \tfrac{1}{2} + \tfrac{-4}{5} \cdot\tfrac{-\sqrt{3}}{2} = \dfrac{-3+4\sqrt{3}}{10} \),\( \cos(A+B) =\dfrac{-4-3\sqrt{3}}{10} \),\( \tan(A+B) =\dfrac{48 - 25\sqrt{3}}{-11} \\ \)
\( \quad \;\; \sin(A-B) =\dfrac{-3-4\sqrt{3}}{10} \),\( \cos(A-B) =\dfrac{-4+3\sqrt{3}}{10} \),\( \tan(A-B) =\dfrac{48 + 25\sqrt{3}}{-11} \)

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

C: SolveEquations

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. \(0.373 + \frac{2\pi}{3} k\) and \(0.674 + \frac{2\pi}{3} k\), 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. \(\dfrac{\tan (x+y)}{1+\tan x \tan y}=\dfrac{\tan x + \tan y}{1-\tan^2 x \tan^2 y}\)

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 odd exercises: 71. \(\tan \left (\frac{x}{10} \right) \)
75.\(\cot\;A ~+~ \cot\;B = \dfrac{\cos(A)}{\sin(A)} + \dfrac{\cos(B)}{\sin(B)} \\
= \dfrac{\cos(A)}{\sin(A)} \cdot\dfrac{\sin(B)}{\sin(B)} + \dfrac{\cos(B)}{\sin(B)} \cdot\dfrac{\sin(A)}{\sin(A)}\\ \)
\( \quad \;\;
=\dfrac{\cos(A)\sin(B) +\cos(B)\sin(A) }{\sin(A) \sin(B) } = \dfrac{\sin\;(A+B)}{\sin\;A~\sin\;B}\)
77. \(\dfrac{\sin(r - s)}{\cos(r)\cos(s)} = \dfrac{\sin (r) \cos (s) - \cos (r) \sin (s)}{\cos(r)\cos(s)}
\\ \)
\( \quad \;\;
=\dfrac{\sin (r) \cos (s)}{\cos(r)\cos(s)} - \dfrac{\cos (r) \sin (s)}{\cos(r)\cos(s)}
=\dfrac{\sin (r) }{\cos(r)} - \dfrac{ \sin (s)}{\cos(s)}= \tan(r) - \tan(s)\)
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. \(\dfrac{\tan (x+y)}{1+\tan x \tan y}= \dfrac{\tan x + \tan y}{1- \tan x \tan y} \cdot \dfrac{1}{1+\tan x \tan y}= \dfrac{\tan x + \tan y}{1-\tan^2 x \tan^2 y}\)
85.\(\cot\;(A-B) = \dfrac{1}{\tan (A-B)}
=\dfrac{1+\tan A \tan B}{\tan A + \tan B}
=\dfrac{1+\tan A \tan B}{\tan A + \tan B} \cdot \dfrac{\cot A \cot B}{\cot A \cot B}
= \dfrac{\cot\;A~\cot\;B \;+\; 1}{\cot\;B \;-\; \cot\;A}\)
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 odd exercises.: 91. \(\cos(x+y)\cos(x-y) =(\cos x \cos y - \sin x \sin y)(\cos x \cos y + \sin x \sin y)
=\cos^2 x \cos^2 y - \sin ^2 x \sin ^2 y \)
\( \quad \;\;=(1-\sin^2 x )(1-\sin^2 y) - \sin ^2 x \sin ^2 y
=1 -\sin^2 x -\sin^2 y+\sin ^2 x \sin ^2 y - \sin ^2 x \sin ^2 y\)
\( \quad \;\;
=1 -\sin^2 x -\sin^2 y= \cos^2x-\sin^2y\)
93. \( \sin(4x)-\sin(3x)\cos x = \sin (x + 3x) -\sin(3x)\cos x\)
\( \quad \;\;
= \sin x \cos (3x) + \cos x \sin (3x) -\sin(3x)\cos x= \sin x \cos(3x) \)
95. \( \sin(3x)\cos(6x) = \sin(3x)\cos(6x) + \cos(3x)\sin(6x) -\cos(3x)\sin(6x)= \sin(9x)-\cos(3x)\sin(6x)\)
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. \( \sin(2x) = \sin x \cos x + \cos x \sin x= 2 \sin x \cos x\)
101. \( \tan(2\theta ) = \dfrac{ \tan \theta + \tan \theta}{1- \tan \theta \tan \theta}= \dfrac{2\tan \theta }{1-\tan^2\theta } \)

6.4e: Exercises - Sum and Difference Identities (2024)

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Key Equations

Sum Formula for Cosine	cos(α+β)=cosαcosβ−sinαsinβ
Sum Formula for Sine	sin(α+β)=sinαcosβ+cosαsinβ
Difference Formula for Sine	sin(α−β)=sinαcosβ−cosαsinβ
Sum Formula for Tangent	tan(α+β)=tanα+tanβ1−tanαtanβ
Difference Formula for Tangent	cos(α−β)=cosαcosβ+sinαsinβ

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