23. Trigonometrical Equations and their General Solutions

23.1. Trigonometrical Equation

An equation involving one or more trigonometrical ratios of unknown angle is called trigonometrical equation.

Ex. $\cos^2x - 4\sin x = 1$

A trigonometrical identity is satisfied for every value of the unknown angle whereas trigonometrical equation is satisfied for only some values of unknown angle. For example, $1 - \cos^2x = \sin^2x$ is a trigonometrical identity because it is satisfied for every value of $x.$

23.2. Solution of a Trigonometrical Equation

A value of the unknown angle which satisfies the given trigonometrical equation is called a solution or root of the equation.

For example, $2\sin\theta = 1 \Rightarrow \theta = 30^\circ, 150^\circ$ which are two solutions between $0$ and $2\pi.$

23.3. General Solution

Some trigonometrical functions are periodic functions, therefore, solutions of trigonometrical equations can be generalized with the help of periodicirty of trigonometrical functions. The solution consisting of all possible solutions of a trigonometrical equation is called its general solution.

For example, $\sin\theta = 0$ has a genral solution which is $n\pi$ where $n\in I.$

Similarly, fo $\cos\theta = 0,$ the general solution is $(2n + 1)\frac{\pi}{2},$ where $n\in I$ and for $\tan\theta = 0$ the solution is again $n\pi.$

23.3.1. General Solution of $\sin\theta = \sin\alpha$

Given, $\sin\theta = \sin\alpha \Rightarrow \sin\theta - \sin\alpha = 0$

$\Rightarrow 2\cos\frac{\theta + \alpha}{2}\sin\frac{\theta - \alpha}{2} = 0$

Case I $\cos\frac{\theta + \alpha}{2} = 0$

$\Rightarrow \theta + \alpha = (2m + 1)\pi, m\in I$

Case II $\sin\frac{\theta - \alpha}{2} = 0$

$\Rightarrow \theta - \alpha = 2m\pi \Rightarrow \theta = 2m\pi + \alpha$

Thus, $\theta = n\pi + (-1)^n\alpha, n\in I$

23.3.2. General Solution of $\cos\theta = \cos\alpha$

Given, $\cos\theta = \cos\alpha \Rightarrow \cos\theta - \cos\alpha = 0$

$\Rightarrow 2\sin\frac{\alpha + \theta}{2}\sin\frac{\theta - \alpha}{2} = 0$

Case I: $\sin\frac{\alpha + \theta}{2} = 0$

$\alpha + \theta = 2n\pi \Rightarrow \theta = 2n\pi - \alpha$

Case II: $\sin\frac{\theta - \alpha}{2} = 0 \Rightarrow \theta = 2n\pi + \alpha$

Thus, $\theta = 2n\pi \pm\alpha$

23.3.3. General Solution of $\tan\theta = \tan\alpha$

Given $\tan\theta = \tan\alpha \Rightarrow \frac{\sin\theta}{\cos\theta} = \frac{\sin\alpha}{\cos\alpha}$

$\Rightarrow \sin(\theta - \alpha) = 0 \therefore \theta - \alpha = n\pi$

$\theta = n\pi + \alpha$

23.4. Principal Value

For any equation having multiple solutions, the solution having least numerical value is known as principal value.

Example: Let $\sin\theta = \frac{1}{2}$ then $\theta = \pi/6, 5\pi/6, 13\pi/6, 17\pi/6, \ldots, -7\pi/6, -11\pi/6, \ldost$

As $\pi/6$ is the least numerical value so it is the principal value in this case.

23.4.1. Method for Finding Principal Value

For this case we consider $\sin\theta = -\frac{1}{2}.$ Since it is negative, $\theta$ will be in third or fourth quadrant. We can approach this either using clockwise direction or annticlockwise direction. If we take anticlockwise direction principal value will be greater than $\pi$ and in case of clockwise direction it will be less than $\pi.$ For principal value, we have to take numerically smallest angle.

So for principal value:

1. If the angle is in 1st or 2nd quadrant we must select anticlockwise direction i.e. principal value will be positive. If the angle is in 3rd or 4th quadrant we must select clockwise direction i.e. principal value will be negative.

2. Principal value is always numerically smaller than $\pi$

3. Principal values always lies in the first circle i.e. first rotation.

23.5. Tips for Finding Complete Solution

1. There should be no extraneous root.

2. There should be no less root.

3. Squaring should be avoided as far as possible. If it is done then check for extraneous roots.

4. Never cancel equal terms containing unknown on two sides which are in product. It may cause root loss.

5. The answer should not contain such values of root which may make any of the terms undefined.

6. Domain should not change. If it changes, necessary correction must be made.

7. Check that denominator is not zero at any stage while solving equations.

23.6. Problems

Find the most general values of $\theta$ satisfying the equations:

1. $\sin\theta = -1$

2. $\cos\theta = -\frac{1}{2}$

3. $\tan\theta = \sqrt{3}$

4. $\sec\theta = -\sqrt{2}$

Solve the equations:

5. $\sin9\theta = \sin\theta$

6. $\sin5x = \cos2x$

7. $\sin3x = \sin x$

8. $\sin3x = \cos2x$

9. $\sin ax + \cos bx = 0$

10. $\tan x\tan4x = 1$

11. $\cos\theta = \sin105^\circ + \cos 105^\circ$

Solve the following:

12. $7\cos^2\theta + 3\sin^2\theta = 4$

13. $3\tan(\theta - 15^\circ) = \tan(\theta + 15^\circ)$

14. $\tan x + \cot x = 2$

15. $\sin^2\theta = \sin^2\alpha$

16. $\tan^2x + \cot^2x = 2$

17. $\tan^2x = 3\cosec^2x - 1$

18. $2\sin^2x + \sin^22x = 2$

19. $7\cos^2x + 3\sin^2x = 4$

20. $2\cos2x + \sqrt{2\sin x} = 2$

21. $8\tan^2\frac{x}{2} = 1 + \sec x$

22. $\cos x\cos2x\cos3x = \frac{1}{4}$

23. $\tan x + \tan2x + \tan3x = 0$

24. $\cot x - \tan x - \cos x + \sin x = 0$

25. $2\sin^2x - 5\sin x\cos x - 8\cos^2x = -2$

26. $(1 - \tan x)(1 + \sin2x) = 1 + \tan x$

27. Solve for x,($-\pi \leq x \leq \pi$), the equation $2(\cos x + \cos2x) + \sin2x(1 + 2\cos x) = 2\sin x$

28. Find all the solutions of the equation $4\cos^2x\sin x - 2\sin^2x = 3\sin x$

29. $2 + 7\tan^2x = 3.25\sec^2x$

30. Find all the values of $x$ for which $\cos 2x + \cos 4x = 2\cos x$

31. $3\tan x + \cot x = 5\cosec x$

32. Find the value of $x$ between $0$ and $2\pi$ for which $2\sin^2x = 3\cos x$

33. Find the solution of $\sin^2x - \cos x = \frac{1}{4}$ in the interbal $0$ to $2\pi.$

34. Solve $3\tan^2x - 2\sin x = 0$

35. Find all values of $x$ satisfying the equation $\sin x + \sin5x = \sin 3x$ between $0$ and $\pi.$

36. $\sin6x = \sin4x - \sin2x$

37. $\cos6x + \cos 4x + \cos 2x + 1 = 0$

38. $\cos x + \cos 2x + \cos 3x = 0$

39. Find the values of $x$ between $0$ and $2\pi,$ for which $\cos3x + \cos2x = \sin\frac{3x}{2} + \sin\frac{x}{2}$

40. $\tan x+ \tan2x + \tan3x = \tan x.\tan2x.\tan3x$

41. $\tan x + \tan 2x + \tan x\tan 2x = 1$

42. $\sin2x + \cos2x + \sin x + \cos x + 1 = 0$

43. $\sin x + \sin 2x + \sin 3x = \cos x + \cos 2x + \cos 3x$

44. $\cos6x + \cos4x = \sin3x + \sin x$

45. $\sec4x - \sec2x = 2$

46. $\cos2x = (\sqrt{2} + 1)\left(\cos x - \frac{1}{\sqrt{2}}\right)$

47. Find all the angles between $-pi$ and $\pi$ for which $5\cos2x + 2\cos^2\frac{x}{2} + 1 = 0$

48. $\cot x - \tan x = \sec x$

49. $1 + \sec x = \cot^2\frac{x}{2}$

50. $\cos3x\cos^3x + \sin3x\sin^3x = 0$

51. $\sin^3x + \sin x\cos x + \cos^3x = 1$

52. Find all the value of $x$ between $0$ and $\frac{\pi}{2},$ for which $\sin 7x + \sin4x + \sin x = 0$

53. $\sin x + \sqrt{3}\cos x = \sqrt{2}$

54. Find the values of $x$ for which $27^{\cos2x}.81^{\sin2x}$ is minimum. Also, find this minimum value.

55. If $32\tan^8x = 2\cos^2y - 3\cos y$ and $3\cos2x = 1,$ then find the general value of $y.$

56. Find all the values of $x$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for which $(1 - \tan x)(1 + \tan x)sec^2x + 2^{\tan^2x} = 0$

57. Solve the equation $e^{\cos x} = e^{-\cos x} + 4.$

58. If $(1 + \tan x)(1 + \tan y) = 2.$ Find all the values of $x + y.$

59. If $\tan(\cot x) = \cot(\tan x),$ prove that $\sin 2x = \frac{4}{(2n + 1)\pi}$

60. If $x$ and $y$ are two distinct roots of the equation $a\tan z+ b\sec z = c.$ Prove that $\tan(x + y) = \frac{2ac}{a^2 - c^2}$

61. If $\sin(\pi\cos x) = \cos(\pi\sin x),$ prove that 1. $\cos\left(x \pm \frac{\pi}{4}\right) = \frac{1}{2\sqrt{2}}$ 2. $\sin2x = -\frac{3}{4}$

62. Determine the smallest positive values of $x$ for which $\tan(x + 100^\circ) = \tan(x + 50^\circ).\tan x.\tan(x - 50^\circ)$

63. Find the general value of $x$ for which $\tan^2x + \sec 2x = 1.$

64. Solve the equation $\sec x - \cosec x = \frac{4}{3}$

65. Find solutions $x\in[0, 2\pi]$ of equation $\sin2x - 12(\sin x - \cos x) + 12 = 0.$

66. Find the smallest positive number r:math:p for which the equation $\cos(p\sin x) = \sin(p\cos x)$ has a solution for $x\in [0, 2\pi].$

67. Solve $\cos x + \sqrt{3}\sin x = 2\cos2x$

68. Solve $\tan x+ \sec x = \sqrt{3}$ for $x\in[0, 2\pi].$

69. Solve $1 + \sin^3x + \cos^3x = \frac{3}{2}\sin2x$

70. Solve the equation $(2 + \sqrt{3})\cos x = 1 - \sin x$

71. Solve the equation $\tan\left(\frac{\pi}{2}\sin x\right) = \cot\left(\frac{\pi}{2}\cos x\right)$

72. Solve $8\cos x\cos2x\cos4x = \frac{\sin6x}{\sin x}$

73. Solve $3 - 2\cos x - 4\sin x -\cos 2x + \sin 2x = 0$

74. Solve $\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x$

75. Solve $\sin^2x\tan x + \cos^2x\cot x - \sin 2x = 1 + \tan x + \cot x$

76. Find the most general value of $x$ which satisfies both the equations $\sin x= -\frac{1}{2}$ and $\tan x = \frac{1}{\sqrt{3}}$

77. If $\tan(x - y) = 1$ and $\sec(x + y) = \frac{2}{\sqrt{3}},$ find the smallest positive values of $x$ and $y$ and their most general value.

78. Find the points of intersection of the curves $y = \cos x$ and $y = \sin 3x$ if $-\frac{\pi}{2}\leq x\leq \frac{\pi}{2}.$

79. Find all values of $x\in [0, 2\pi]$ such that $r\sin x = \sqrt{3}$ and $r + 4\sin x = 2(\sqrt{3} + 1)$

80. Find the smallest positive values of $x$ and $y$ satisfying $x - y = \frac{\pi}{4}$ and $\cot x + \cot y = 2.$

81. Find the general values of $x$ and $y$ such that $5\sin x\cos y = 1$ and $4\tan x = \tan y.$

82. Find all values of $x$ lying between $0$ and $2\pi,$ such that $r\sin x = 3$ and $r = 4(1 + \sin x)$

83. If $\sin x = \sin y$ and $\cos x = \cos y$ then prove that either $x = y$ or $x - y = 2n\pi,$ where $n\in I.$

84. If $\cos(x - y) = \frac{1}{2}$ and $\sin(x + y) = \frac{1}{2}$ find the smallest positive values of $x$ and $y$ and also their most general values.

85. Find the points of intersection of the curves $y = \cos 2x$ and $y = \sin x$ for, $-\frac{\pi}{2}\leq x\leq \frac{\pi}{2}.$

86. Find the most general value of $x$ which satisfies the equations $\cos x = \frac{1}{\sqrt{2}}$ and $\tan x = -1.$

87. Find the most general value of $x$ which satisfies the equations $\tan x = \sqrt{3}$ and $\cosec x = -\frac{2}{\sqrt{3}}$

88. If $x$ and $y$ be two distinct values of $z$ lying between $0$ and $2\pi,$ satisfying the equation $3\cos z + 4\sin z = 2,$ find the value of $\sin(x + y).$

89. Show that the equation $2\cos^2\frac{x}{2}\sin^2x = x^2+ x^{-2}$ for $0<x\leq\frac{\pi}{2}$ has no real solution.

90. Find the real value of $x$ such that $y = \frac{3 + 2i\sin x}{1 - 2i\sin x}$ is either real or purely imaginary.

91. Determine for which values of $a$ the equation $a^2 - 2a + \sec^2\pi(a + x) = 0$ has solutions and find them.

92. Find the values of $x$ in $(-\pi, \pi)$ which satisfy the equation $8^{1 + |\cos x| + \cos^2x + |\cos^3 x| + \ldots \text{~to~}\infty} = 4^3$

93. Solve $|\cos x|^{\sin^2x - \frac{3}{2}\sin x + \frac{1}{2}} = 1.$

94. Solve $3^{\sin2x + 2\cos^2x} + 3^{1 -\sin2x + 2\sin^2x} = 28.$

95. If $A = (x/2\cos^2x + \sin x\leq 2)$ and $B = \left(x/\frac{\pi}{2}\leq x\leq \frac{3\pi}{2}\right)$ find $A\cap B$

96. Solve $\sin x + \cos x = 1 + \sin x\cos x.$

97. Solve $\sin6x + \cos4x + 2 = 0.$

98. Prove that the equation $\sin2x + \sin3x + \ldots + \sin nx = n - 1$ has n solution for any arbitrary integer $n>2.$

99. Solve $\cos^7x + \sin^4x = 1.$

100. Find the number of solutions of the equation $\sin x + 2\sin2x = 3 + \sin3x$ in the interval $0\leq x\leq \pi.$

101. For what value of $k$ the equation $\sin x + \cos(k + x) + \cos(k - x) = 2$ has real solutions.

102. Solve for $x$ and $y,$ the equation $x\cos^3y + 3x\cos y.\sin^2y = 14$ and $x\sin^3y + 3x\cos^2y\sin y = 13$

103. Find all the values of $\alpha$ for which the equation $\sin^4x + \cos^4x + \sin2x + \alpha = 0$ is valid.

104. Solve $\tan\left(x + \frac{\pi}{4}\right) = 2\cot x - 1.$

105. If $x, y$ be two angles both satisfying the equation $a\cos 2z + b\sin2z = c,$ prove that $\cos^2x + \cos^2y = \frac{a^2 + ac + b^2}{a^2 + b^2}$

106. If $x_1, x_2, x_3, x_4$ be roots of the equation $\sin(x + y) = k\sin 2x,$ no two of which differ by a multiple of $2\pi,$ prove that $x_1 + x_2 + x_3 + x_4 = (2n + 1)\pi.$

107. Show that the equation $\sec x + \cosec x = c$ has two roots between $0$ and $\pi$ if $c^2<8$ and four roots if $c^2 > 8.$

108. Let $\lambda$ and $\alpha$ be real. Find the set of all values of $\lambda$ for which the system of linear equations $\lambda x + y\sin\alpha + z\cos\alpha = 0, x + y\cos\alpha + z\sin\alpha = 0, -x + y\sin\alpha - z\cos\alpha = 0$ has non-trivial solution. For $\lambda = 1,$ find all the values of $\alpha.$

109. Find the values of $x$ and $y, 0<x,y<\frac{\pi}{2},$ satisfying the equation $\cos x \cos y\cos(x + y) = -\frac{1}{8}$

110. Find the number of distinct real roots of $\begin{vmatrix}\sin x& \cos x & \cos x \\\cos x & \sin x & \cos x\\\cos x & \cos x & \sin x\end{vmatrix} = 0$ in the interval $-\frac{\pi}{4}\leq x\leq \frac{\pi}{4}.$

111. Find the number of values of $x$ in the interval $[0, 5\pi]$ satisfying the equation $3\sin^2x - 7\sin x + 2 = 0.$

112. Find the range of $y$ such that the following equation in $x,$ $y + \cos x = \sin x$ has a real solution. For $y = 1,$ find $x$ such that $0\leq x\leq2\pi.$

113. Solve $\sum_{r = 1}^n\sin(rx)\sin(r^2x) = 1$

114. Show that the equation $\sin x(\sin x + \cos x) = a$ has real solutions if $a$ is a real number lying between $\frac{1}{2}(1 - \sqrt{2})$ and $\frac{1}{2}(1 + \sqrt{2}).$

115. Find the real solutions of the equation $2\cos^2\frac{x^2 + x}{6} = 2^x + 2^{-x}.$

116. Solve the inequality $\sin x\geq \cos2x.$

117. Find the general solution of the equation $\left(\cos\frac{x}{4} - 2\sin x\right)\sin x + \left(1 + \sin \frac{x}{4} -2\cos x\right)\cos x = 0$

118. Find the general solution of the equation $2(\sin x -\cos2x) - \sin2x(1 + 2\sin x) + 2\cos x = 0.$

119. Solve $\frac{\sin2x}{\sin\frac{2x + \pi}{3}} = 0.$

120. Solve the equation $3\tan2x - 4\tan3x = \tan^23x\tan2x$

121. Solve the equation $\sqrt{1 + \sin2x} = \sqrt{2}\cos2x.$

122. Show that $x = 0$ is the only solution satisfying the equation $1 + \sin^2ax = \cos x$ where $a$ is irrational.

123. Consider the system of linear equarions in $x, y$ and $z, x\sin3\theta -y + z = 0, x\cos2\theta + 4y + 3z = 0, 2x + 7y + 7z = 0.$ Find the values of $\theta$ for which the system has non-trivial solutions.

124. Find all the solutions of the equation $\sin x + \sin\frac{\pi}{8}\sqrt{(1 - \cos x)^2 + \sin^2x} = 0$ in the interval $\left[\frac{5\pi}{2}, \frac{7\pi}{2}\right]$

125. Let $A = \{x: \tan x -\tan^2x > 0\}$ and $y = \left\{x: |\sin x|<\frac{1}{2}\right\}$. Determine $A\cap B.$

126. If $0\leq x\leq 2\pi,$ then solve $2^{\frac{1}{\sin^2x}}\sqrt{y^2 - 2y + 2}\leq 2$

127. If $|\tan x| = \tan x + \frac{1}{\cos x}(0\leq x\leq 2\pi)$ then prove that $x = \frac{7\pi}{6}$ or $\frac{11\pi}{6}$

128. Find the smallest positive solution satisfying $\log_{\cos x}\sin x + \log_{\sin x}\cos x = 2$

129. Solve the inequality $\sin x\cos x + \frac{1}{2}\tan x\geq 1$

130. Solve $\tan x^{\cos^2 x} = \cot x^{\sin x}$

131. If $0\leq \alpha, \beta \leq 3,$ then $x^2 + 4 + 3\cos(\alpha x + \beta) = 2x$ has at least one solution, then prove thatt $\alpha + \beta = \pi, 3\pi.$

132. Prove that the equation $2\sin x = |x| + a$ has no solution for $a\in \left(\frac{3\sqrt{3 - \pi}}{3}, \infty\right)$