Trigonometric Equation
1. (a) Define trigonometric equation with example.
Solution:
An equation involving trigonometric ratios is called trigonometric equation.
For example: (i) \( \cos \theta = \frac{\sqrt{3}}{2} \) (ii) \( \sin^2 \beta = \frac{1}{2} \)
(b) What do you mean by root (solution) of the given trigonometric equation?
Solution:
The value of an unknown angle which satisfies the given trigonometric equation is known as its root or solution.
2. (a) If \( \sec \theta = -2 \) , what is the least positive angle in \( 1^{st} \) quadrant.
Solution:
In first quadrant all the trigonometric ratios are positive, so there is not any angle in the first quadrant that satisfies \( \sec \theta = -2 \) .
2. (b) How to find the angle in \( 4^{th} \) quadrant, if the least positive angle \( \theta \) is given.
Solution:
If the least positive angle \( \theta \) is given, then the angle in \( 4^{th} \) quadrant \( = 360^{\circ} - \theta \)
2. (c) What are the minimum and maximum values of \( \sin \theta \) and \( \cos \theta \)
Solution:
The minimum value of \( \sin \theta \) and \( \cos \theta \) \( = -1 \)
The maximum values of \( \sin \theta \) and \( \cos \theta \) \( = 1 \)
3. Solve: (a) \( \sin \theta = \frac{\sqrt{3}}{2} \) \( ( 0^{\circ} \le \theta \le 90^{\circ} ) \)
Solution:
\begin{align} \sin \theta & = \frac{\sqrt{3}}{2} \\ \text{or, } \sin \theta & = \sin 60^{\circ} \\ \therefore & \theta = 60^{\circ} \\ \end{align}
(b) \( \cos \theta = \frac{ 1 }{2} \) \( ( 0^{\circ} \le \theta \le 90^{\circ} ) \)
Solution:
\begin{align} \cos \theta & = \frac{1 }{2} \\ \text{or, } \cos \theta & = \cos 60^{\circ} \\ \therefore \ \ & \theta = 60^{\circ} \\ \end{align}
(c) \( \sqrt{3}\cot \theta = 1 \) \( ( 0^{\circ} \le \theta \le 90^{\circ} ) \)
Solution:
\begin{align} \sqrt{3}\cot \theta & = 1 \\ \text{or, } \cot \theta & = \frac{1}{\sqrt{3}} \\ \text{or, } \tan \theta & = \sqrt{3} \\ \text{or, } \tan \theta & = \tan 60^{\circ} \\ \therefore \ \ \theta & = 60^{\circ} \\ \end{align}
(d) \( \tan \theta - 1 = 0 \) \( ( 0^{\circ} \le \theta \le 90^{\circ} ) \)
Solution:
\begin{align} \tan \theta - 1 & = 0 \\ \text{or, } \tan \theta & = 1 \\ \text{or, } \tan \theta & = \tan 45^{\circ} \\ \therefore \ \ \theta & = 45^{\circ} \\ \end{align}
(e) \( 2\sin \theta - 1 = 0 \) \( ( 0^{\circ} \le \theta \le 90^{\circ} ) \)
Solution:
\begin{align} 2\sin \theta - 1 & = 0 \\ \text{or, } 2 \sin \theta & = 1 \\ \text{or, } \sin \theta & = \frac{1}{2} \\ \text{or, } \sin \theta & = \sin 30^{\circ} \\ \therefore \ \ \theta & = 30^{\circ} \\ \end{align}
(f ) \( \sin \theta = 1\) \( ( 0^{\circ} \le \theta \le 90^{\circ} ) \)
Solution:
\begin{align} \sin \theta & = 1 \\ \text{or, } \sin \theta & = \sin 90^{\circ} \\ \therefore \ \ \theta & = 90^{\circ} \\ \end{align}
(g) \( \cos \theta - \frac{1}{\sqrt{2}} = 0 \) \( ( 0^{\circ} \le \theta \le 90^{\circ} ) \)
Solution:
\begin{align} \cos \theta - \frac{1}{\sqrt{2}} & = 0 \\ \text{or, } \cos \theta & = \frac{1}{\sqrt{2}} \\ \text{or, } \cos \theta & = \cos 45^{\circ} \\ \therefore \ \ \theta & = 45^{\circ} \\ \end{align}
(h) \( \sec \theta = 2 \) \( ( 0^{\circ} \le \theta \le 90^{\circ} ) \)
Solution:
\begin{align} \sec \theta & = 2 \\ \text{or, } \cos \theta & = \frac{1}{2} \\ \text{or, } \cos \theta & = \cos 60^{\circ} \\ \therefore \ \ \theta & = 60^{\circ} \\ \end{align}
4. Solve: (a) \( 2\cos \theta + 1 = 0 \) \( ( 0^{\circ} \le \theta \le 180^{\circ} ) \)
Solution:
\begin{align} 2\cos \theta + 1 & = 0 \\ \text{or, } 2 \cos \theta & = -1 \\ \text{or, } \cos \theta & = \frac{-1}{2} \\ \text{or, } \cos \theta & = \cos ( 180^{\circ} -60^{\circ} ) \\ \text{or, } \cos \theta & = \cos 150^{\circ} \\ \therefore \ \ \theta & = 150^{\circ} \\ \end{align}
(b) \( \sqrt{2} \sec \theta + 2 = 0 \) \( ( 0^{\circ} \le \theta \le 180^{\circ} ) \)
Solution:
\begin{align} \sqrt{2} \sec \theta + 2 & = 0 \\ \text{or, } \sqrt{2} \sec \theta & = -2 \\ \text{or, } \sec \theta & = \frac{-2}{\sqrt{2} } \\ \text{or, } \cos \theta & = - \frac{\sqrt{2}}{2} \\ \text{or, } \cos \theta & = -\frac{\sqrt{2}}{\sqrt{2} \times \sqrt{2} } \\ \text{or, } \cos \theta & = -\frac{1 }{ \sqrt{2} }\\ \text{or, } \cos \theta & = \cos (180^{\circ} - 45^{\circ}) \\ \therefore \ \ \theta & = 135^{\circ} \\ \end{align}
(c) \( 2\sin \theta - \sqrt{3} = 0 \) \( ( 0^{\circ} \le \le 180^{\circ} ) \)
Solution:
\begin{align} 2\sin \theta - \sqrt{3} & = 0 \\ \text{or, } 2\sin \theta & = \sqrt{3} \\ \text{or, } \sin \theta & = \frac{\sqrt{3}}{2} \\ \text{or, } \sin \theta & = \sin 60^{\circ}, \sin (180^\circ -60^\circ ) \\ \therefore \ \ \theta & = 60^{\circ}, 120^\circ \\ \end{align}
(d) \( 3\cot \theta - \sqrt{3} = 0 \) \( ( 0^{\circ} \le \alpha, \theta \le 180^{\circ} ) \)
Solution:
\begin{align} 3\cot \theta - \sqrt{3} & = 0 \\ \text{or, } 3\cot \theta & = \sqrt{3} \\ \text{or, } \cot \theta & = \frac{\sqrt{3}}{3} \\ \text{or, } \cot \theta & = \frac{\sqrt{3}}{\sqrt{3} \times \sqrt{3} } \\ \text{or, } \cot \theta & = \frac{1}{ \sqrt{3} } \\ \text{or, } \tan \theta & = \sqrt{3} \\ \text{or, } \tan \theta & = \tan 60^{\circ} \\ \therefore \ \ \theta & = 60^{\circ} \\ \end{align}
(e) \( \sqrt{3} \text{cosec} \theta - 2 = 0 \) \( ( 0^{\circ} \le \alpha, \theta \le 180^{\circ} ) \)
Solution:
\begin{align} \ & \sqrt{3} \text{ cosec} \theta - 2 = 0 \\ \text{or, } & \sqrt{3} \text{cosec} \theta = 2 \\ \text{or, } & \text{cosec} \theta = \frac{2}{\sqrt{3}} \\ \text{or, } & \sin \theta = \frac{\sqrt{3}}{2} \\ \text{or, } & \sin \theta = \sin 60^{\circ} , \sin ( 180^{\circ} -60^{\circ} ) \\ \text{or, } & \sin \theta = \sin 60^{\circ}, \sin 120^{\circ} \\ \therefore \ \ & \theta = 60^{\circ}, 120^{\circ} \\ \end{align}
(f) \( \sqrt{3} \tan \theta + 1 = 0 \) \( ( 0^{\circ} \le \alpha, \theta \le 180^{\circ} ) \)
Solution:
\begin{align} \ & \ \sqrt{3} \tan \theta + 1 = 0 \\ \text{or, } & \sqrt{3} \tan \theta = - 1 \\ \text{or, } & \tan \theta = \frac{-1}{\sqrt{3}} \\ \text{or, } & \tan \theta = \tan (180^{\circ} - 30^{\circ} ) \\ \text{or, } & \tan \theta = \tan 150^{\circ} \\ \therefore \ \ & \theta = 150^{\circ} \\ \end{align}
5. (a) Solve: \( 2\sin^2 \alpha -1 = 0 \) \( [ 0^{\circ} \le \alpha \le 180^{\circ} ] \)
Solution:
\begin{align} 2\sin^2 \alpha -1 = 0 \\ \text{or, } 2\sin^2 \alpha & = 1 \\ \text{or, } \sin^2 \alpha & = \frac{1}{2} \\ \text{or, } \sin \alpha & = \pm \sqrt{\frac{1}{2}} \\ \text{or, } \sin \alpha & = \pm \frac{1}{\sqrt{2}} \\ \end{align}
Now,
| Taking positive | Taking negative |
|---|---|
| \( \sin \alpha = \frac{1}{\sqrt{2}} \) | \( \sin \alpha = \frac{-1}{\sqrt{2}} \) |
| or, sin \(\alpha \) = sin 45°, sin (180° - 45°) | Rejected |
| or, \(\alpha \) = 45°, 135° |
\( \therefore \alpha = 45^{\circ}, 135^{\circ} \)
5. (b) Solve: \( 4\sin \alpha = 3 \text{cosec} \alpha \) \( [ 0^{\circ} \le \alpha \le 180^{\circ} ] \)
Solution:
\begin{align} 4\sin \alpha & = 3 \text{cosec} \alpha \\ \text{or, } 4 \sin \alpha & = 3\times \frac{1}{\sin \alpha } \\ \text{or, } 4 \sin^2 \alpha & = 3 \\ \text{or, } \sin^2 \alpha & = \frac{3 }{4} \\ \text{or, } \sin \alpha & = \pm \sqrt{\frac{3}{4}} \\ \text{or, } \sin \alpha & = \pm \frac{\sqrt{3}}{2} \\ \end{align}
Now,
| Taking positive | Taking negative |
|---|---|
| or, sin α = √3/2 | sin α = -√3/2 |
| or, sin α = sin 60°, sin (180° - 60°) | Rejected |
| or, α = 60°, 120° |
5. (c) Solve: \( \tan^2 \alpha - 1 = 2 \) \( [ 0^{\circ} \le \alpha \le 180^{\circ} ] \)
Solution: \begin{align} \tan^2 \alpha - 1 & = 2 \\ \text{or, } \tan^2 \alpha & = 2 +1 \\ \text{or, } \tan^2 \alpha & = 3 \\ \text{or, } \tan \alpha & = \pm \sqrt{3} \\ \end{align} Now,
| Taking positive | Taking negative |
|---|---|
| tan α = √3 | tan α = -√3 |
| or, tan α = tan 60° | or, tan α = tan (180° - 60°) |
| or, α = 60° | or, α = 120° |
Solution: \begin{align} 3 \cot^2 \theta - 1 = 0 \\ \text{or, } 3 \cot^2 \theta = 1 \\ \text{or, } \cot^2 \theta = \frac{1}{3} \\ \text{or, } \tan^2 \theta = 3 \\ \text{or, } \tan \theta = \pm \sqrt{3} \\ \end{align} Now,
| Taking positive | Taking negative |
|---|---|
| tan α = √3 | tan α = -√3 |
| or, tan α = tan 60° | or, tan α = tan (180° - 60°) |
| or, α = 60° | or, α = 120° |
5. (e) Solve: \( 4\sin^2 \alpha = \tan^2 60^{\circ} \) \( [ 0^{\circ} \le \alpha \le 360^{\circ} ] \)
Solution: \begin{align} 4\sin^2 \alpha & = \tan^2 60^{\circ} \\ \text{or, } 4\sin^2 \alpha & = (\sqrt{3} )^2 \\ \text{or, } 4\sin^2 \alpha & = 3 \\ \text{or, } \sin^2 \alpha & = \frac{3}{4} \\ \text{or, } \sin \alpha & = \pm \sqrt{\frac{3}{4}} \\ \text{or, } \sin \alpha & = \pm \frac{\sqrt{3}}{2} \\ \end{align} Now,
| Taking positive | Taking negative |
|---|---|
| sin α = √3/2 | sin α = -√3/2 |
| sin α = sin 60°, sin (180° - 60°) | sin α = sin (180°+60°), sin (360°-60°) |
| α = 60°, 120° | α = 240°, 300° |
5. (f) Solve: \( \sqrt{3} \tan \theta + 1 = 0 \) \( [ 0^{\circ} \le \alpha \le 180^{\circ} ] \)
Solution: \begin{align} & \sqrt{3} \tan \theta + 1 = 0 \\ \text{or, } & \sqrt{3} \tan \theta = - 1 \\ \text{or, } & \tan \theta = \frac{-1}{\sqrt{3}} \\ \text{or, } & \tan \theta = \tan ( 180 ^{\circ} - 60^{\circ} ) \\ \text{or, } & \theta = 180 ^{\circ} - 60^{\circ} \\ \therefore \ \ & \ \ \theta = 120 ^{\circ} \\ \end{align}
6. (a) Solve: \( 2\cos^2 \theta = - \sqrt{3} \cos \theta \) \( (0^{\circ}\le \theta \le 180^{\circ} ) \)
Solution: \begin{align} \ & 2\cos^2 \theta = - \sqrt{3} \cos \theta \\ \text{or, } & 2\cos^2 \theta + \sqrt{3} \cos \theta = 0 \\ \text{or, } & \cos \theta ( 2\cos \theta + \sqrt{3} ) = 0 \\ \end{align} Now,
| Either | Or |
|---|---|
| \(\cos \theta = 0\) | 2\(\cos \theta + \sqrt{3} = 0\) |
| or, \(\cos \theta = \cos 90^{\circ}\) | or, 2\(\cos \theta = - \sqrt{3}\) |
| or, \(\theta = 90^{\circ}\) | or, \(\cos \theta = \frac{-\sqrt{3}}{2}\) |
| or, \(\cos \theta = \cos (180^{\circ} - 30^{\circ})\) | |
| or, \(\cos \theta = \cos 150^{\circ}\) | |
| or, \(\theta = 150^{\circ}\) |
6. (b) Solve: \( 2\cos^2 \theta =3\sin \theta \) \( (0^{\circ}\le \theta \le 180^{\circ} ) \)
Solution: \begin{align} \ & 2\cos^2 \theta =3\sin \theta \\ \text{or, } & 2(1- \sin^2 \theta) = 3 \sin \theta \\ \text{or, } & 2 - 2\sin^2 \theta = 3 \sin \theta \\ \text{or, } & 0 = 2\sin^2 \theta + 3 \sin \theta - 2 \\ \text{or, } & 2\sin^2 \theta + 3 \sin \theta - 2 = 0 \\ \text{or, } & 2\sin^2 \theta + 4 \sin \theta -\sin \theta - 2 = 0 \\ \text{or, } & 2\sin \theta ( \sin \theta + 2 ) - 1( \sin \theta + 2 ) = 0 \\ \text{or, } & ( \sin \theta + 2 )( 2\sin \theta - 1 )= 0 \\ \end{align} Now,
| Either | Or |
|---|---|
| \( \sin \theta + 2 = 0 \) | \( 2\sin \theta - 1 = 0 \) |
| or, \( \sin \theta = -2 \) | or, \(2\sin \theta = 1 \) |
| Rejected | or, \( \sin \theta = \frac{1}{2} \) |
| or, \( \sin \theta = \sin 30^{\circ}, \sin (180^{\circ} - 30^{\circ}) \) | |
| or, \( \sin \theta = \sin 30^{\circ}, \sin 150^{\circ} \) | |
| or, \( \theta = 30^{\circ}, 150^{\circ} \) |
6. (c) Solve: \( \text{cosec} \theta - 2\sin \theta = 1 \) \( (0^{\circ}\le \theta \le 180^{\circ} ) \)
Solution: \begin{align} \ & \text{cosec} \theta - 2\sin \theta = 1 \\ \text{or, } & \frac{1}{\sin \theta } - 2\sin \theta = 1 \\ \text{or, } & \frac{1-2\sin^2 \theta}{\sin \theta } = 1\\ \text{or, } & 1-2\sin^2 \theta = \sin \theta \\ \text{or, } & 0 = 2\sin^2 \theta + \sin \theta -1 \\ \text{or, } & 2\sin^2 \theta + \sin \theta -1 = 0 \\ \text{or, } & 2\sin^2 \theta + 2 \sin \theta -\sin \theta - 1 = 0 \\ \text{or, } & 2\sin \theta ( \sin \theta + 1 ) - 1( \sin \theta + 1 ) = 0 \\ \text{or, } & ( \sin \theta + 1 )( 2\sin \theta - 1 )= 0 \\ \end{align} Now,
$$\begin{array}{|l|l|} \hline \text{either } & \text{ or } \\ \sin \theta +1 = 0 & 2\sin \theta - 1 \\ \text{or, } \sin \theta = - 1 & \text{or, } 2\sin \theta = 1 \\ \text{rejected } & \text{or, } \sin \theta = \frac{1}{2} \\ \hline \end{array}$$
\( \therefore \ \ \theta = 30^{\circ}, 150^{\circ} \)6. (d) Solve: \( \cos^2 \frac{\theta}{2} -\cos \frac{\theta}{2} + \frac{1}{4} = 0 \) \( (0^{\circ}\le \theta \le 180^{\circ} ) \)
Solution: \begin{align} \ & \cos^2 \frac{\theta}{2} -\cos \frac{\theta}{2} + \frac{1}{4} = 0 \\ \text{or, } & \frac{4 \cos^2 \frac{\theta}{2} - 4 \cos \frac{\theta}{2} + 1 }{4} = 0 \\ \text{or, } & 4 \cos^2 \frac{\theta}{2} - 4 \cos \frac{\theta}{2} + 1 =0 \\ \text{or, } & \left(2\cos \frac{\theta}{2} - 1 \right)^2 = 0 \\ \text{or, } & 2\cos \frac{\theta}{2} - 1 =0 \\ \text{or, } & 2\cos \frac{\theta}{2} = 1 \\ \text{or, } & \cos \frac{\theta}{2} = \frac{1}{2} \\ \text{or, } & \cos \frac{\theta}{2} = \cos 60^{\circ} \\ \text{or, } & \frac{\theta}{2} = 60^{\circ} \\ \text{or, } & \theta = 2\times 60^{\circ} \\ \therefore \ \ \ & \theta = 120^{\circ} \\ \end{align}
6. (e) Solve: \( \cos \theta ( 2\sin \theta - 1 ) = 0 \) \( (0^{\circ}\le \theta \le 180^{\circ} ) \)
Solution: \( \cos \theta ( 2\sin \theta - 1 ) = 0 \) Now,
| Either | Or |
|---|---|
| \( \cos \theta = 0 \) | \( 2\sin \theta - 1 = 0 \) |
| or, \( \cos \theta = \cos 90^{\circ} \) | or, \( 2\sin \theta = 1 \) |
| or, \( \theta = 90^{\circ} \) | or, \( \sin \theta = \frac{1}{2} \) |
| or, \( \sin \theta = \sin 30^{\circ}, \sin (180^{\circ} - 30^{\circ})\) | |
| or, \( \theta = 30^{\circ}, 150^{\circ} \) |
6. (f) Solve: \( \sin 2\theta = \sin \theta \) \( (0^{\circ}\le \theta \le 180^{\circ} ) \)
Solution: \begin{align} \ & \sin 2\theta = \sin \theta \\ \text{or, } & 2\sin \theta \cos \theta = \sin \theta \\ \text{or, } & 2\sin \theta \cos \theta - \sin \theta =0 \\ \text{or, } & \sin \theta (2 \cos \theta - 1)=0 \\ \end{align} Now,
| Either | Or |
|---|---|
| \(\sin \theta = 0\) | \(2\cos \theta - 1 = 0\) |
| or, \(\sin \theta = \sin 0^{\circ}, \sin 180^\circ \) | or, \(2\cos \theta = 1\) |
| or, \(\theta = 0^{\circ}, 180^\circ \) | or, \(\cos \theta = \frac{1}{2}\) |
| or, \(\cos \theta = \cos 60^{\circ} \) | |
| or, \(\theta = 60^{\circ} \) |
6. (g) Solve: \( \sin 3\theta = \cos 6\theta \) \( (0^{\circ}\le \theta \le 180^{\circ} ) \)
Solution: \begin{align} \sin 3\theta & = \cos 6\theta \\ \text{or, } \sin 3\theta & = \sin (90^{\circ} - 6\theta ) \sin (5\times 90^{\circ} - 6\theta ), \sin (9\times 90^{\circ} - 6\theta ) , \sin (13\times 90^{\circ} - 6\theta ), \sin (17\times 90^{\circ} - 6\theta ) \\ \text{or, } 3\theta & = 90^{\circ} - 6\theta, 5\times 90^{\circ} - 6\theta, 9\times 90^{\circ} - 6\theta,13\times 90^{\circ} - 6\theta,17\times 90^{\circ} - 6\theta \\ \text{or, } 3\theta + 6\theta & = 90^{\circ} , 5\times 90^{\circ}, 9\times 90^{\circ} ,13\times 90^{\circ},17\times 90^{\circ}\\ \text{or, } 9\theta & = 90^{\circ} , 5\times 90^{\circ}, 9\times 90^{\circ} ,13\times 90^{\circ},17\times 90^{\circ}\\ \text{or, } 9\theta & = 90^{\circ} , 5\times 90^{\circ}, 9\times 90^{\circ} ,13\times 90^{\circ},17\times 90^{\circ}\\ \text{or, } \theta & = \frac{90^{\circ}}{9} , \frac{5\times 90^{\circ}}{9}, \frac{9\times 90^{\circ}}{9} ,\frac{13\times 90^{\circ}}{9},\frac{17\times 90^{\circ}}{9}\\ \text{or, } \theta & = 10^{\circ}, 50^{\circ}, 90^{\circ}, 130^{\circ}, 170^{\circ} \\ \end{align}
6. (h) Solve: \( \cot 5\theta = \tan \theta \) \( (0^{\circ}\le \theta \le 180^{\circ} ) \)
Solution: \begin{align} \cot 5\theta & = \tan \theta \\ \text{or, } \cot 5\theta & = \cot (90^{\circ} - \theta ), \cot (3\times 90^{\circ} - \theta ), \cot (5\times 90^{\circ} - \theta ) , \cot (7\times 90^{\circ} - \theta ), \cot (9\times 90^{\circ} - \theta ), \cot (11\times 90^{\circ} - \theta ) \\ \text{or, } 5 \theta & = 90^{\circ} - \theta, 3\times 90^{\circ} - \theta, 5\times 90^{\circ} - \theta,7\times 90^{\circ} - \theta,9\times 90^{\circ} - \theta, 11\times 90^{\circ} - \theta \\ \text{or, } 5\theta + \theta & = 90^{\circ} , 3\times 90^{\circ}, 5\times 90^{\circ} ,7\times 90^{\circ},9\times 90^{\circ},11\times 90^{\circ} - \theta\\ \text{or, } 6\theta & = 90^{\circ} , 3\times 90^{\circ}, 5\times 90^{\circ} ,7\times 90^{\circ},9\times 90^{\circ},11\times 90^{\circ} \\ \text{or, } 6\theta & = 90^{\circ} , 3\times 90^{\circ}, 5\times 90^{\circ} ,7\times 90^{\circ},9\times 90^{\circ},11\times 90^{\circ}\\ \text{or, } \theta & = \frac{90^{\circ}}{6} , \frac{3\times 90^{\circ}}{6}, \frac{5\times 90^{\circ}}{6} ,\frac{7\times 90^{\circ}}{6},\frac{9\times 90^{\circ}}{6}, \frac{11\times 90^{\circ}}{6}\\ \text{or, } \theta & = 15^{\circ}, 45^{\circ}, 75^{\circ},105^{\circ},135^{\circ},165^{\circ} \\ \end{align}
7. Solve:- \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
(a) \( 3\sin^2 x + 4\cos x = 4 \)
Solution: \begin{align} \ & 3\sin^2 x + 4\cos x = 4 \\ \text{or, } & 3(1-\cos^2 x ) + 4\cos x = 4 \\ \text{or, } & 3 - 3\cos^2 x + 4\cos x = 4 \\ \text{or, } & 0 = 3\cos^2 x - 4\cos x + 4 - 3 \\ \text{or, } & 3\cos^2 x - 4\cos x + 1 = 0 \\ \text{or, } & 3\cos^2 x - (3+1)\cos x + 1 = 0 \\ \text{or, } & 3\cos^2 x - 3\cos x - \cos x+ 1 = 0 \\ \text{or, } & 3\cos x ( \cos x - 1 ) - 1 ( \cos x - 1 ) = 0 \\ \text{or, } & (\cos x - 1 )(3\cos x - 1 ) = 0 \\ \end{align}
| Either | Or |
|---|---|
| \(\cos x - 1 = 0\) | \(3\cos x - 1 = 0\) |
| or, \(\cos x = 1\) | or, \(3\cos x = 1\) |
| or, \(\cos x = \cos 0^{\circ}, \cos 360^{\circ}\) | or, \(\cos x = \frac{1}{3}\) |
| or, \(x = 0^{\circ}, 360^{\circ}\) | or, \(x = \cos^{-1}\left(\frac{1}{3}\right)\) |
7. Solve:- (b) \( \cos^2 x = 3\sin^2 x + 4\cos x \) \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
Solution: \begin{align} \ & \cos^2 x = 3\sin^2 x + 4\cos x \\ \text{or, } & \cos^2 x = 3( 1 - \cos^2 x ) + 4\cos x \\ \text{or, } & \cos^2 x = 3 - 3\cos^2 x + 4\cos x \\ \text{or, } & \cos^2 x + 3\cos^2 x - 4\cos x - 3 = 0\\ \text{or, } & 4\cos^2 x - 4\cos x - 3 = 0 \\ \text{or, } & 4\cos^2 x - (6-2)\cos x - 3 = 0 \\ \text{or, } & 4\cos^2 x - 6\cos x + 2\cos x - 3 = 0 \\ \text{or, } & 2\cos x (2\cos x - 3) + 1 (2\cos x - 3) = 0 \\ \text{or, } & (2\cos x - 3) (2\cos x + 1 ) = 0 \\ \end{align}
| Either | Or |
|---|---|
| \(2\cos x - 3 = 0\) | \(2\cos x + 1 = 0\) |
| or, \(2\cos x = 3\) | or, \(2\cos x = - 1\) |
| or, \( \cos x = \frac{3}{2} \) | or, \(\cos x = \frac{-1}{2}\) |
| Rejected | or, \(\cos x = \cos (180^\circ - 60^{\circ}), \cos(180^{\circ} + 60^{\circ})\) |
| or, \(\cos x = \cos 120^{\circ}, \cos 240^{\circ}\) | |
| or, \(x = 120^{\circ}, 240^{\circ}\) |
7. Solve:- (c) \( \tan x + \cot x = 2 \) \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
Solution: \begin{align} \ & \tan x + \cot x = 2 \\ \text{or, } & \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x } = 2 \\ \text{or, } & \frac{\sin^2 x + \cos^2 x }{\cos x \sin x } = 2 \\ \text{or, } & \frac{1}{\cos x \sin x } = 2 \\ \text{or, } & \frac{1}{\cos x \sin x } = 2 \\ \text{or, } & 1= 2 \sin x \cos x \\ \text{or, } & 2\sin x \cos x = 1 \\ \text{or, } & \sin 2x = 1 \\ \text{or, } & \sin 2x = \sin 90^{\circ}, \sin 5\times 90^{\circ} \\ \text{or, } & 2x = 90^{\circ}, 5\times 90^{\circ} \\ \text{or, } & x = \frac{90^{\circ}}{2}, \frac{5\times 90^{\circ}}{2} \\ \therefore \ \ & x = 45^{\circ}, 225^{\circ} \\ \end{align}
7. Solve:- (d) \( \tan x -\sin x = 0 \) \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
Solution: \begin{align} \ & \tan x -\sin x = 0 \\ \text{or, } & \frac{\sin x }{\cos x} - \sin x = 0 \\ \text{or, } & \frac{\sin x - \sin x \cos x }{\cos x} = 0 \\ \text{or, } & \sin x - \sin x \cos x = 0 \\ \text{or, } & \sin x ( 1 - \cos x ) = 0 \\ \end{align}
| Either | Or |
|---|---|
| \(\sin x = 0\) | \(1 - \cos x = 0\) |
| or, \(\sin x = \sin 90^{\circ}\) | or, \(\cos x = 1\) |
| or, \(x = 90^{\circ}\) | or, \(\cos x = \cos 0^{\circ}\) |
| or, \(\cos x = \cos 0^{\circ}, \cos(360^{\circ} - 0^{\circ})\) | |
| or, \(\cos x = \cos 0^{\circ}, \cos 360^{\circ}\) | |
| or, \(x = 0^{\circ}, 360^{\circ}\) |
7. Solve:- \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
(e) \( \sec x . \tan x = \sqrt{2} \)
Solution: \begin{align} \ & \sec x . \tan x = \sqrt{2} \\ \text{or, } & \frac{ 1 }{\cos x} . \frac{\sin x }{\cos x } =\sqrt{2} \\ \text{or, } & \frac{\sin x}{\cos^2 x } = \sqrt{2} \\ \text{or, } & \sin x = \sqrt{2} \cos^2 x \\ \text{or, } & \sin x = \sqrt{2} (1-\sin^2 x ) \\ \text{or, } & \sin x = \sqrt{2} - \sqrt{2} \sin^2 x \\ \text{or, } & \sqrt{2} \sin^2 x +\sin x - \sqrt{2} = 0 \\ \text{or, } & \sqrt{2} \sin^2 x +(2-1) \sin x - \sqrt{2} = 0 \\ \text{or, } & \sqrt{2} \sin^2 x +2 \sin x - \sin x - \sqrt{2} = 0 \\ \text{or, } & \sqrt{2} \sin x ( \sin x +\sqrt{2} ) - 1 (\sin x + \sqrt{2}) = 0 \\ \text{or, } & ( \sin x +\sqrt{2} ) (\sqrt{2} \sin x - 1 ) = 0 \\ \end{align}
| Either | Or |
|---|---|
| \(\sin x + \sqrt{2} = 0\) | \(\sqrt{2} \sin x - 1 = 0\) |
| or, \(\sin x = - \sqrt{2}\) | or, \(\sqrt{2} \sin x = 1\) |
| Rejected | or, \(\sin x = \frac{1}{\sqrt{2}}\) |
| or, \(\sin x = \sin 45^{\circ}, \sin (180^{\circ} - 45^{\circ})\) | |
| or, \(\sin x = \sin 45^{\circ}, \sin 135^{\circ}\) | |
| or, \(x = 45^{\circ}, 135^{\circ}\) |
7. Solve: (f) \( \cot^2 x + \text{cosec}^2x = 3 \) \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
Solution: \begin{align} \ & \cot^2 x + \text{cosec}^2x = 3 \\ \text{or, } & \cot^2 x + 1+\cot^2 x = 3\\ \text{or, } & 2\cot^2 x = 3 - 1 \\ \text{or, } & 2\cot^2 x = 2 \\ \text{or, } & \cot^2 x = 1 \\ \text{or, } & \cot x = \pm \sqrt{ 1} \\ \text{or, } & \cot x = \pm 1 \\ \end{align}
| Taking positive | Taking negative |
|---|---|
| cot x = 1 | cot x = -1 |
| or, cot x = cot 45°, cot(180° + 45°) | or, cot x = cot(180° - 45°), cot(360° - 45°) |
| or, cot x = cot 45°, cot 225° | or, cot x = cot 135°, cot 315° |
| or, x = 45°, 225° | x = 135°, 315° |
7. Solve: (g) \( (1-\sqrt{3} ) \tan x + 1 + \sqrt{3} = \sqrt{3} \sec^2 \theta \) \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
Solution: \begin{align} \ & (1-\sqrt{3} ) \tan x + 1 + \sqrt{3} = \sqrt{3} \sec^2 x \\ \text{or, } & \tan x - \sqrt{3} \tan x + 1 + \sqrt{3} = \sqrt{3} (1+\tan^2 x) \\ \text{or, } & \tan x - \sqrt{3} \tan x + 1 + \sqrt{3} = \sqrt{3} + \sqrt{3} \tan^2 x \\ \text{or, } & 0 = \sqrt{3} \tan^2 x+ \sqrt{3} \tan x - \tan x - 1 \\ \text{or, } & \sqrt{3} \tan^2 x+ \sqrt{3} \tan x - \tan x - 1 = 0 \\ \text{or, } & \sqrt{3} \tan x (\tan x + 1 ) - 1(\tan x + 1) = 0 \\ \text{or, } & (\tan x + 1 ) (\sqrt{3} \tan x - 1 ) = 0 \\ \end{align}
| Either | Or |
|---|---|
| \( \tan x + 1 = 0 \) | \( \sqrt{3} \tan x - 1 = 0 \) |
| or, \( \tan x = -1 \) | or, \( \sqrt{3} \tan x = 1 \) |
| or, \( \tan x = \tan (180° - 45°), \tan (360° - 45°) \) | or, \( \tan x = \frac{1}{\sqrt{3}} \) |
| or,\( \tan x = \tan 135°, \tan 315° \) | or, \( \tan x = \tan 30° , \tan (180° + 30°) \) |
| or,\( x = 135°, 315° \) | or, \( x = 30°, 210° \) |
7. Solve: (h) \( \cot^2 x + \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right) \cot x + 1 = 0 \) \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
Solution: \begin{align} \ & \cot^2 x + \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right) \cot x + 1 = 0 \\ \text{or, } & \cot^2 x + \sqrt{3} \cot x + \frac{1}{\sqrt{3}} \cot x + 1 = 0 \\ \text{or, } & \cot x \left( \cot x + \sqrt{3} \right) + \frac{ 1}{\sqrt{3}} \left( \cot x + \sqrt{3} \right) = 0 \\ \text{or, } & \left( \cot x + \sqrt{3} \right) \left( \cot x + \frac{1}{\sqrt{3}} \right) = 0 \\ \end{align}
| Either | Or |
|---|---|
| \( \cot x + \sqrt{3} = 0 \) | \( \cot x + \frac{1}{\sqrt{3}} = 0 \) |
| or, \( \cot x = -\sqrt{3} \) | or, \( \cot x = - \frac{1}{\sqrt{3}} \) |
| or, \( \cot x = \cot (180° - 30°), \cot (360° - 30°) \) | or, \( \cot x = \cot (180° - 60°), \cot (360° - 60°) \) |
| or, \( \cot x = \cot 150°, \cot 330° \) | or, \( \cot x = \cot 120°, \cot 300° \) |
| or, \( x = 150°, 330° \) | or, \( x = 120°, 300° \) |
7. Solve: (i) \( \tan^2 x + \left( 1- \sqrt{3} \right) \tan x - \sqrt{3} = 0 \) \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
Solution: \begin{align} \ & \tan^2 x + \left( 1- \sqrt{3} \right) \tan x - \sqrt{3} = 0 \\ \text{or, } & \tan^2 x + \tan x - \sqrt{3} \tan x - \sqrt{3} = 0 \\ \text{or, } & \tan x ( \tan x + 1 ) - \sqrt{3} ( \tan x + 1 ) = 0 \\ \text{or, } & (\tan x + 1 ) ( \tan x - \sqrt{3} )= 0 \\ \end{align}
| Either | Or |
|---|---|
| \( \tan x + 1 = 0 \) | \( \tan x - \sqrt{3} = 0 \) |
| or, \( \tan x = -1 \) | or, \( \tan x = \sqrt{3} \) |
| or, \( \tan x = \tan (180° - 45°), \tan (360° - 45°) \) | or,\( \tan x = \tan 60°, \tan (180° + 60°) \) |
| or, \( \tan x = \tan 135°, \tan 315° \) | or, \( \tan x = \tan 60°, \tan 240° \) |
| or, \( x = 135°, 315° \) | or, \( x = 60°, 240° \) |
7. Solve: (j) \( 2\sin x + \cot x - \text{cosec} x = 0 \) \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
Solution: \begin{align} \ & 2\sin x + \cot x - \text{cosec} x = 0 \\ \text{or, } & 2\sin x + \frac{\cos x }{\sin x} -\frac{1}{ \sin x } = 0 \\ \text{or, } & \frac{ 2\sin^2 x + \cos x - 1 }{\sin x } = 0 \\ \text{or, } & 2\sin^2 x + \cos x - 1 = 0 \\ \text{or, } & 2(1-\cos^2 x )+ \cos x - 1 = 0 \\ \text{or, } & 2-2\cos^2 x + \cos x - 1 = 0 \\ \text{or, } & -2\cos^2 x + \cos x +1 = 0 \\ \text{or, } & 2\cos^2 x - \cos x - 1 = 0 \\ \text{or, } & 2\cos^2 x - (2-1) \cos x - 1 = 0 \\ \text{or, } & 2\cos^2 x - 2 \cos x + \cos x - 1 = 0 \\ \text{or, } & 2\cos x (\cos x - 1 ) + 1 ( \cos x - 1 ) = 0 \\ \text{or, } & ( \cos x - 1 ) (2\cos x + 1 )= 0 \\ \end{align}
| Either | Or |
|---|---|
| \( \cos x - 1 = 0 \) | \( 2cos x + 1 = 0 \) |
| or,\( \cos x = 1 \) | or, \( 2 \cos x = -1 \) |
| or,\( cos x = cos 0°, cos 360° \) | or, \( \cos x = \frac{-1}{2} \) |
| or,\( x = 0°, 360° \) | or, \( \cos x = \cos (180° - 60°), \cos (180° + 60°) \) |
| or, \( x = 120°, 240° \) |
8. Solve: (a) \( \sqrt{3} \sin \theta + \cos \theta = 1 \) \( ( 0^{\circ} \le \theta \le 360^{\circ} ) \)
Solution: \( \sqrt{3} \sin \theta + \cos \theta = 1 ...(i) \) \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin \theta )^2 + (\text{ coeff. of } \cos \theta )^2 } \\ \ & = \sqrt{ ( \sqrt{3} )^2 + 1^2 }\\ & = \sqrt{3+1}\\ & = \sqrt{4}\\ & = 2 \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } 2\\ \ & \text{We get, }\\ \ & \frac{\sqrt{3}}{2} \sin \theta + \frac{1}{2} \cos \theta = \frac{1}{2} \\ \text{or, } & \sin \theta \times \frac{\sqrt{3}}{2} + \cos \theta \times \frac{1}{2} = \frac{1}{2} \\ \text{or, } & \sin \theta \cos 30^{\circ} + \cos \theta \sin 30^{\circ} = \frac{1}{2} \\ \text{or, } & \sin ( \theta + 30^{\circ} ) = \sin 30^{\circ}, \sin ( 180^{\circ} - 30^{\circ} ), \sin ( 360^{\circ}+ 30^{\circ} ) \\ \text{or, } & \sin ( \theta + 30^{\circ} ) = \sin 30^{\circ}, \sin 150^{\circ}, \sin 390^{\circ} \\ \text{or, } & \theta + 30^{\circ} = 30^{\circ}, 150^{\circ}, 390^{\circ} \\ \text{or, } & \theta = 30^{\circ} - 30^{\circ}, 150^{\circ}- 30^{\circ}, 390^{\circ}- 30^{\circ} \\ \therefore \ \ & \theta = 0^{\circ}, 120^{\circ} ,360^{\circ}\\ \end{align}
8. Solve: (b) \( \sin \theta + \sqrt{3} \cos \theta = 1 \) \( ( 0^{\circ} \le \theta \le 360^{\circ} ) \)
Solution: \( \sin \theta + \sqrt{3} \cos \theta = 1 ...(i) \) \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin \theta )^2 + (\text{ coeff. of } \cos \theta )^2 } \\ \ & = \sqrt{ 1^2+( \sqrt{3} )^2 }\\ & = \sqrt{1+3}\\ & = \sqrt{4}\\ & = 2 \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } 2\\ \ & \text{We get, }\\ \ & \frac{1}{2} \sin \theta + \frac{\sqrt{3}}{2} \cos \theta = \frac{1}{2} \\ \text{or, } & \sin \theta \times \frac{1}{2} + \cos \theta \times \frac{\sqrt{3}}{2} = \frac{1}{2} \\ \text{or, } & \sin \theta \cos 60^{\circ} + \cos \theta \sin 60^{\circ} = \frac{1}{2} \\ \text{or, } & \sin ( \theta + 60^{\circ} ) = \sin 30^{\circ}, \sin ( 180^{\circ} - 30^{\circ} ), \sin ( 360^{\circ} + 30^{\circ} ) \\ \text{or, } & \sin ( \theta + 60^{\circ} ) = \sin 30^{\circ}, \sin 150^{\circ}, \sin 390^{\circ} \\ \text{or, } & \theta + 60^{\circ} = 30^{\circ}, 150^{\circ}, 390^{\circ} \\ \text{or, } & \theta = 30^{\circ} - 60^{\circ}, 150^{\circ}- 60^{\circ}, 390^{\circ}- 60^{\circ} \\ \text{or, } & \theta = -30^{\circ}\text{ (Rejected) }, 90^{\circ}, 330^{\circ} \\ \ \therefore & \theta = 90^{\circ}, 330^{\circ} \\ \end{align}
8. Solve: (c) \( \sin \theta + \cos \theta = \sqrt{2} \) \( ( 0^{\circ} \le \theta \le 360^{\circ} ) \)
Solution: \( \sin \theta + \cos \theta = \sqrt{2} ...(i) \) \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin \theta )^2 + (\text{ coeff. of } \cos \theta )^2 } \\ \ & = \sqrt{ 1^2+ 1^2 }\\ & = \sqrt{1+1}\\ & = \sqrt{2}\\ \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } \sqrt{2} \\ \ & \text{We get, }\\ \ & \frac{1}{\sqrt{2}} \sin \theta + \frac{1}{\sqrt{2}} \cos \theta = \frac{\sqrt{2}}{\sqrt{2}} \\ \text{or, } & \sin \theta \times \frac{1}{\sqrt{2}} + \cos \theta \times \frac{1}{\sqrt{2}} = 1 \\ \text{or, } & \sin \theta \cos 45^{\circ} + \cos \theta \sin 45^{\circ} = 1 \\ \text{or, } & \sin ( \theta + 45^{\circ} ) =\sin 90^{\circ} \\ \text{or, } & \theta + 45^{\circ} = 90^{\circ} \\ \text{or, } & \theta = 90^{\circ} - 45^{\circ} \\ \therefore \ \ & \theta = 45^{\circ} \\ \end{align}
8. Solve: (d) \( \cos \theta + \frac{1}{\sqrt{3}} \sin \theta = 1 \) \( ( 0^{\circ} \le \theta \le 360^{\circ} ) \)
Solution: \begin{align} \ & \cos \theta + \frac{1}{\sqrt{3}} \sin \theta = 1 \\ \ & \text{or,} \frac{\sqrt{3} \cos \theta + \sin \theta}{\sqrt{3}}= 1 \\ \ & \text{or, } \sqrt{3} \cos \theta + \sin \theta = \sqrt{3} \\ & \text{or, } \sin \theta + \sqrt{3} \cos \theta = \sqrt{3} ... (i) \end{align} \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin \theta )^2 + (\text{ coeff. of } \cos \theta )^2 } \\ \ & = \sqrt{ 1^2+( \sqrt{3} )^2 }\\ & = \sqrt{1+3}\\ & = \sqrt{4}\\ & = 2 \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } 2\\ \ & \text{We get, }\\ \ & \frac{1}{2} \sin \theta + \frac{\sqrt{3}}{2} \cos \theta = \frac{\sqrt{3}}{2} \\ \text{or, } & \sin \theta \times \frac{1}{2} + \cos \theta \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \\ \text{or, } & \sin \theta \cos 60^{\circ} + \cos \theta \sin 60^{\circ} = \sin 60^{\circ}, \sin ( 180^{\circ} - 60^{\circ} ) ,\sin ( 360^{\circ} + 60^{\circ} ) \\ \text{or, } & \sin ( \theta + 60^{\circ} ) = \sin 60^{\circ}, \sin 120^{\circ}, \sin 420^{\circ}\\ \text{or, } & \theta + 60^{\circ} = 60^{\circ},120^{\circ}, 420^{\circ} \\ \text{or, } & \theta = 60^{\circ} - 60^{\circ}, 120^{\circ}-60,420^{\circ}-60^{\circ} \\ \therefore \ \ & \theta = 0^{\circ},60^{\circ},360^{\circ} \\ \end{align}
8. Solve: (e) \( \sin \theta + \cos \theta = \frac{1}{\sqrt{2}} \) \( ( 0^{\circ} \le \theta \le 360^{\circ} ) \)
Solution: \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin \theta )^2 + (\text{ coeff. of } \cos \theta )^2 } \\ \ & = \sqrt{ 1^2+1^2 }\\ & = \sqrt{1+1}\\ & = \sqrt{2} \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } \sqrt{2} \\ \ & \text{We get, }\\ \ & \frac{1}{\sqrt{2}} \sin \theta + \frac{1}{\sqrt{2}} \cos \theta = \frac{\frac{1}{\sqrt{2}}}{\sqrt{2}} \\ \text{or, } & \sin \theta \times \frac{1}{\sqrt{2}} + \cos \theta \times \frac{1}{\sqrt{2}} = \frac{1}{2} \\ \text{or, } & \sin \theta \cos 45^{\circ} + \cos \theta \sin 45^{\circ} = \sin 30^{\circ}, \sin ( 180^{\circ} - 30^{\circ} ) ,\sin ( 360^{\circ} + 30^{\circ} ) \\ \text{or, } & \sin ( \theta + 45^{\circ} ) = \sin 30^{\circ}, \sin 150^{\circ}, \sin 390^{\circ}\\ \text{or, } & \theta + 45^{\circ} = 30^{\circ},150^{\circ}, 390^{\circ} \\ \text{or, } & \theta = 30^{\circ} - 45^{\circ}, 150^{\circ}-45^{\circ},390^{\circ}-45^{\circ} \\ \text{or, } & \theta = -15^{\circ}\text{ (Refected) },105^{\circ},345^{\circ} \\ \therefore \ \ & \theta = 105^{\circ},345^{\circ} \\ \end{align}
8. Solve: (f) \( \cos x + \sqrt{3} \sin x = 2 \) \( ( 0^{\circ} \le x \le 360^{\circ} ) \)
Solution: \( \cos x + \sqrt{3} \sin x = 2 \)
\( \text{or, } \sqrt{3} \sin x + \cos x = 2 \) \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin x )^2 + (\text{ coeff. of } \cos x )^2 } \\ \ & = \sqrt{ ( \sqrt{3} )^2 + 1^2 }\\ & = \sqrt{3+1}\\ & = \sqrt{4}\\ & = 2 \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } 2\\ \ & \text{We get, }\\ \ & \frac{\sqrt{3}}{2} \sin x + \frac{1}{2} \cos x = \frac{2}{2} \\ \text{or, } & \sin x \times \frac{\sqrt{3}}{2} + \cos x \times \frac{1}{2} = 1 \\ \text{or, } & \sin x \cos 30^{\circ} + \cos x \sin 30^{\circ} = 1 \\ \text{or, } & \sin ( x + 30^{\circ} ) = \sin 90^{\circ}\\ \text{or, } & x+ 30^{\circ} = 90^{\circ} \\ \text{or, } & x = 90^{\circ} - 30^{\circ} \\ \therefore \ \ & x = 60^{\circ}\\ \end{align}
8. Solve: (g) \( \tan \theta + \sqrt{3} \sec \theta = \sqrt{3} \) \( ( 0^{\circ} \le \theta \le 360^{\circ} ) \)
Solution: \begin{align} & \tan \theta + \sqrt{3} \sec \theta = \sqrt{3} \\ \text{or, } & \frac{\sin \theta }{\cos \theta} + \sqrt{3} \times \frac{1}{\cos \theta } = \sqrt{3} \\ \text{or, } & \frac{ \sin \theta + \sqrt{3} }{\cos \theta } = \sqrt{3} \\ \text{or, } & \sin \theta + \sqrt{3} = \sqrt{3} \cos \theta \\ \text{or, } & \sin \theta - \sqrt{3} \cos \theta = -\sqrt{3} .......(i) \end{align} \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin \theta )^2 + (\text{ coeff. of } \cos \theta )^2 } \\ \ & = \sqrt{1^2+ ( \sqrt{3} )^2 }\\ & = \sqrt{1+3}\\ & = \sqrt{4}\\ & = 2 \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } 2\\ \ & \text{We get, }\\ \ & \frac{1}{2} \sin \theta - \frac{\sqrt{3}}{2} \cos \theta = \frac{-\sqrt{3}}{2} \\ \text{or, } & \sin \theta \times \frac{1}{2} - \cos \theta \times \frac{\sqrt{3}}{2} = \frac{-\sqrt{3}}{2} \\ \text{or, } & \sin \theta \cos 60^{\circ} - \cos \theta \sin 60^{\circ} = \frac{-\sqrt{3}}{2} \\ \text{or, } & \sin ( \theta - 60^{\circ} ) = \sin (180^{\circ}+60^{\circ}), \sin (360^{\circ}-60^{\circ} )\\ \text{or, } & \theta- 60^{\circ} = 240^{\circ}, 300^{\circ} \\ \text{or, } & \theta = 240^{\circ} + 60^{\circ}, 300^{\circ}+60^{\circ} \\ \therefore \ \ & \theta = 300^{\circ}, 360^{\circ}\\ \end{align}
8. Solve:- (h) \( \text{cosec}\theta + \cot \theta = \sqrt{3} \) \( ( 0^{\circ} \le \theta \le 360^{\circ} ) \)
Solution: \begin{align} & \text{cosec}\theta + \cot \theta = \sqrt{3} \\ \text{or, } & \frac{1 }{\sin \theta} + \frac{\cos \theta }{\sin \theta } = \sqrt{3} \\ \text{or, } & \frac{ 1 + \cos \theta }{\sin \theta } = \sqrt{3} \\ \text{or, } & 1 + \cos \theta = \sqrt{3} \sin \theta \\ \text{or, } & 1 = \sqrt{3} \sin \theta - \cos \theta \\ \text{or, } & \sqrt{3} \sin \theta - \cos \theta = 1 ........(i) \end{align} \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin \theta )^2 + (\text{ coeff. of } \cos \theta )^2 } \\ \ & = \sqrt{ (- \sqrt{3} )^2 +1^2 }\\ & = \sqrt{3+1}\\ & = \sqrt{4}\\ & = 2 \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } 2\\ \ & \text{We get, }\\ \ & \frac{\sqrt{3}}{2} \sin \theta - \frac{1}{2} \cos \theta = \frac{1}{2} \\ \text{or, } & \sin \theta \times \frac{\sqrt{3}}{2} - \cos \theta \times \frac{1}{2} = \frac{1}{2} \\ \text{or, } & \sin \theta \cos 30^{\circ} - \cos \theta \sin 30^{\circ} = \frac{1}{2} \\ \text{or, } & \sin ( \theta - 30^{\circ} ) = \sin 30^{\circ}, \sin (180^{\circ}-30^{\circ}), \sin (360^{\circ} + 30^{\circ} )\\ \text{or, } & \theta- 30^{\circ} = 30^{\circ}, 150^{\circ}, 390^{\circ} \\ \text{or, } & \theta = 30^{\circ}+ 30^{\circ}, 150^{\circ} + 30^{\circ}, 390^{\circ} + 30^{\circ} \\ \text{or, } & \theta = 60^{\circ}, 180^{\circ}, 420^{\circ} \text{ (Rejected) }\\ \therefore \ \ & \theta = 60^{\circ}, 180^{\circ} \\ \end{align}
8. Solve:- (i) \( \sqrt{3} \tan \theta + 1 = \sec \theta \) \( ( 0^{\circ} \le \theta \le 360^{\circ} ) \)
Solution: \begin{align} & \sqrt{3} \tan \theta + 1 = \sec \theta \\ \text{or, } & \sqrt{3} \times \frac{\sin \theta }{\cos \theta} + 1 = \frac{1}{\cos \theta } \\ \text{or, } & \frac{ \sqrt{3} \sin \theta + \cos \theta }{\cos \theta } = \frac{1}{\cos \theta } \\ \text{or, } & \sqrt{3} \sin \theta + \cos \theta = 1 ........(i) \end{align} \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin \theta )^2 + (\text{ coeff. of } \cos \theta )^2 } \\ \ & = \sqrt{ (\sqrt{3} )^2 +1^2 }\\ & = \sqrt{3+1}\\ & = \sqrt{4}\\ & = 2 \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } 2\\ \ & \text{We get, }\\ \ & \frac{\sqrt{3}}{2} \sin \theta - \frac{1}{2} \cos \theta = \frac{1}{2} \\ \text{or, } & \sin \theta \times \frac{\sqrt{3}}{2} + \cos \theta \times \frac{1}{2} = \frac{1}{2} \\ \text{or, } & \sin \theta \cos 30^{\circ} + \cos \theta \sin 30^{\circ} = \frac{1}{2} \\ \text{or, } & \sin ( \theta + 30^{\circ} ) = \sin 30^{\circ}, \sin (180^{\circ}-30^{\circ}), \sin (360^{\circ} + 30^{\circ} )\\ \text{or, } & \theta + 30^{\circ} = 30^{\circ}, 150^{\circ}, 390^{\circ} \\ \text{or, } & \theta = 30^{\circ} - 30^{\circ}, 150^{\circ} - 30^{\circ}, 390^{\circ} - 30^{\circ} \\ \therefore \ \ & \theta =0^{\circ}, 120^{\circ}, 360^{\circ} \\ \end{align}
9. Solve:- (a) \( \sin 4\theta + \sin 2\theta = 0 \) \( ( 0^{\circ} \le \theta \le 180^{\circ} ) \)
Solution: \begin{align} & \sin 4\theta + \sin 2\theta = 0 \\ \text{or, } & 2\sin \left(\frac{4\theta + 2\theta }{2} \right)\cos \left(\frac{4\theta - 2\theta }{2} \right) = 0 \\ \text{or, } & 2\sin \left(\frac{6\theta }{2} \right)\cos \left(\frac{2\theta }{2} \right) = 0 \\ \text{or, } & 2\sin 3\theta \cos \theta = 0 \\ \text{or, } & \sin 3\theta \cos \theta = 0 \end{align}
| Either | Or |
|---|---|
| \( \sin 3θ = 0 \) | \( \cos θ = 0 \) |
| or, \( \sin 3θ = \sin 0^\circ, \sin 180^\circ, \sin 360^\circ, \sin 540^\circ \) | or, \( \cos θ = \cos 90^\circ \) |
| or, \( 3θ = 0^\circ, 180^\circ, 360^\circ, 540^\circ \) | or, \( θ = 90^\circ\) |
| or, \( θ = \frac{0^\circ}{3}, \frac{180^\circ}{3}, \frac{360^\circ}{3}, \frac{540^\circ}{3} \) | |
| or, \( θ = 0^\circ, 60^\circ, 120^\circ, 180^\circ \) |
9. Solve: (b) \( \sin 3\theta + \sin 2\theta = \sin \theta \) \( ( 0^{\circ} \le \theta \le 180^{\circ} ) \)
Solution: \begin{align} \ & \sin 3\theta + \sin 2\theta = \sin \theta \\ \text{or, } & \sin 3\theta - \sin \theta + \sin 2\theta = 0 \\ \text{or, } & 2\cos \left(\frac{3\theta + \theta }{2} \right)\sin \left(\frac{3\theta - \theta }{2} \right) + \sin 2\theta =0 \\ \text{or, } & 2\cos \left(\frac{4\theta }{2} \right)\sin \left(\frac{2\theta }{2} \right) + \sin 2\theta = 0 \\ \text{or, } & 2\cos 2\theta \sin \theta + \sin 2\theta = 0 \\ \text{or, } & 2\cos 2\theta \sin \theta + 2\sin \theta \cos \theta = 0 \\ \text{or, } & 2\sin \theta (\cos 2\theta + \cos \theta ) = 0 \\ \end{align}
| Either | Or |
|---|---|
| \( 2 \sin θ = 0 \) | \( \cos 2\theta = - \cos \theta \) |
| or, \( \sin θ = sin 0^\circ, \sin 180^\circ, \sin 360^\circ \) | or,\( \cos 2\theta = \cos (180^\circ - \theta ), \cos ( 180^\circ + \theta ) \) |
| or, \( \theta = 0^\circ, 180^\circ, 360^\circ \) | or, \( 2\theta = 180^\circ - \theta, 180^\circ + \theta \) |
| or, \( 3\theta = 180^\circ \), \( \theta = 180^\circ \) | |
| or, \( \theta = 60^\circ, 180^\circ \) |
9. Solve:- (c) \( \cos 3\theta + \cos \theta = \cos 2\theta \) \( ( 0^{\circ} \le \theta \le 180^{\circ} ) \)
Solution: \begin{align} & \cos 3\theta + \cos \theta = \cos 2\theta \\ \text{or, } & 2\cos \left(\frac{3\theta + \theta }{2} \right)\cos \left(\frac{3\theta - \theta }{2} \right) =\cos 2\theta \\ \text{or, } & 2\cos \left(\frac{4\theta }{2} \right)\cos \left(\frac{2\theta }{2} \right) =\cos 2\theta \\ \text{or, } & 2\cos 2\theta \cos \theta =\cos 2\theta \\ \text{or, } & 2\cos 2\theta \cos \theta - \cos 2\theta = 0 \\ \text{or, } & \cos 2\theta (2\cos \theta - 1) = 0 \\ \end{align}
| Either | Or |
|---|---|
| cos 2θ = 0 | 2cos θ - 1 = 0 |
| or, cos 2θ = cos 90°, cos (360° - 90°) | or, 2cos θ = 1 |
| or, 2θ = 90°, 270° | or, cos θ = \(\frac{1}{2} \) |
| or, θ = \( \frac{90°}{2} \) , \( \frac{270°}{2} \) | or, cos θ = cos 60° |
| or, θ = 45°, 135° | or, θ = 60° |
9. Solve:- (d) \( \cos \theta + \cos 3\theta =- \cos 5\theta \) \( ( 0^{\circ} \le \theta \le 90^{\circ} ) \)
Solution: \begin{align} & \cos \theta + \cos 3\theta =- \cos 5\theta \\ \text{or, } & \cos 5\theta + \cos 3\theta + \cos \theta = 0 \\ \text{or, } & 2\cos \left(\frac{5\theta + 3\theta }{2} \right)\cos \left(\frac{5\theta - 3\theta }{2} \right) +\cos \theta = 0 \\ \text{or, } & 2\cos \left(\frac{8\theta }{2} \right)\cos \left(\frac{2\theta }{2} \right) +\cos \theta = 0 \\ \text{or, } & 2\cos 4\theta \cos \theta +\cos \theta = 0 \\ \text{or, } & \cos \theta (2\cos 4\theta + 1) = 0 \\ \end{align}
| Either | Or |
|---|---|
| cos θ = 0 | 2cos 4θ + 1 = 0 |
| or, cos θ = cos 90° | or, 2cos 4θ = -1 or, cos 4θ = \( \frac{-1}{2} \) |
| or, θ = 90° | or, cos 4θ = cos (180° - 60°), cos (180° + 60°) |
| or, 4θ = 120°, 240° | |
| or, θ = \(\frac{120^\circ}{4} \), \( \frac{240^\circ}{4}\) | |
| or, θ = 30°, 60° | |
9. Solve:- (e) \( \cos 3\theta + \cos \theta = 2\cos \theta \) \( ( 0^{\circ} \le \theta \le 180^{\circ} ) \)
Solution: \begin{align} & \cos 3\theta + \cos \theta = 2\cos \theta \\ \text{or, } & 2\cos \left(\frac{3\theta + \theta }{2} \right)\cos \left(\frac{3\theta - \theta }{2} \right) = 2\cos \theta \\ \text{or, } & 2\cos \left(\frac{4\theta }{2} \right)\cos \left(\frac{2\theta }{2} \right) = 2\cos \theta \\ \text{or, } & 2\cos 2\theta \cos \theta - 2\cos \theta = 0 \\ \text{or, } & 2\cos \theta (\cos 2\theta - 1) = 0 \\ \text{or, } & \cos \theta (\cos 2\theta - 1) = 0 \end{align}
| Either | Or |
|---|---|
| cos θ = cos 90° | cos 2θ - 1 = 0 |
| or, θ = 90° | or, cos 2θ = 1 |
| or, cos 2θ = cos 0°, cos 360° | |
| or, 2θ = 0°, 360° | |
| or, θ = \( \frac{0°}{2} \), \( \frac{360°}{2} \) | |
| or, θ = 0°, 180° |
9. Solve:- (f) \( \sin 2\theta + \sin 4\theta = \cos \theta + \cos 3\theta \) \( ( 0^{\circ} \le \theta \le 180^{\circ} ) \)
Solution: \begin{align} \ & \sin 2\theta + \sin 4\theta = \cos \theta + \cos 3\theta \\ \text{or, } & \sin 4\theta + \sin 2\theta = \cos 3\theta + \cos \theta \\ \text{or, } & 2\sin \left(\frac{4\theta + 2\theta }{2} \right)\cos \left(\frac{4\theta - 2\theta }{2} \right) = 2\cos \left(\frac{3\theta + \theta }{2} \right)\cos \left(\frac{3\theta - \theta }{2} \right) \\ \text{or, } & 2\sin \left(\frac{6\theta }{2} \right)\cos \left(\frac{2\theta }{2} \right) = 2\cos \left(\frac{4\theta }{2} \right)\cos \left(\frac{2\theta }{2} \right) \\ \text{or, } & 2\sin 3\theta \cos \theta = 2\cos 2\theta \cos \theta \\ \text{or, } & \sin 3\theta \cos \theta = \cos 2\theta \cos \theta \\ \text{or, } & \sin 3\theta \cos \theta - \cos 2\theta \cos \theta = 0 \\ \text{or, } & \cos \theta (\sin 3\theta - \cos 2\theta ) = 0 \\ \end{align}
| Either | Or |
|---|---|
| cos θ = 0 | sin 3θ - cos 2θ = 0 |
| or, cos θ = cos 90° | or, sin 3θ = cos 2θ |
| or, θ = 90° | or, sin 3θ = sin (90° - 2θ), sin (5 × 90° - 2θ) |
| or, 3θ + 2θ = 90°, 5 × 90° | |
| or, 5θ = 90°, 5 × 90° | |
| or, θ = 90°/5, 5 × 90°/5 | |
| or, θ = 18°, 90° |
9. Solve:- (g) \( \cos \theta + \sin \theta = \cos 2\theta + \sin 2\theta \) \( ( 0^{\circ} \le \theta \le 180^{\circ} ) \)
Solution: \begin{align} & \cos \theta + \sin \theta = \cos 2\theta + \sin 2\theta \\ \text{or, } & \cos \theta - \cos 2\theta = \sin 2\theta - \sin \theta \\ \text{or, } & 2\sin \left(\frac{\theta + 2\theta }{2} \right)\sin \left(\frac{2\theta - \theta }{2} \right) = 2\cos \left(\frac{2\theta + \theta }{2} \right)\sin \left(\frac{2\theta - \theta }{2} \right) \\ \text{or, } & 2\sin \left(\frac{3\theta }{2} \right)\sin \left(\frac{\theta }{2} \right) = 2\cos \left(\frac{3\theta }{2} \right)\sin \left(\frac{\theta }{2} \right) \\ \text{or, } & \sin \left(\frac{3\theta }{2} \right)\sin \left(\frac{\theta }{2} \right) = \cos \left(\frac{3\theta }{2} \right)\sin \left(\frac{\theta }{2} \right) \\ \text{or, } & \sin \left(\frac{3\theta }{2} \right)\sin \left(\frac{\theta }{2} \right) - \cos \left(\frac{3\theta }{2} \right)\sin \left(\frac{\theta }{2} \right) = 0 \\ \text{or, } & \sin \frac{\theta}{2} \left[ \sin\frac{3\theta}{2} -\cos \frac{3\theta}{2} \right] = 0 \\ \end{align}
| Either | Or |
|---|---|
| sin θ/2 = 0 | sin 3θ/2 - cos 3θ/2 = 0 |
| or, sin θ/2 = sin 0° | or, sin 3θ/2 = cos 3θ/2 |
| or, θ/2 = 0° | or, sin 3θ/2 = sin (90° - 3θ/2), sin (5 × 90° - 3θ/2) |
| or, θ = 0° | or, 3θ/2 = 90° - 3θ/2, 5 × 90° - 3θ/2 |
| or, 3θ/2 + 3θ/2 = 90°, 5 × 90° | |
| or, 6θ/2 = 90°, 5 × 90° | |
| or, 3θ = 90°, 5 × 90° | |
| or, θ = 90°/3, 5 × 90°/3 | |
| or, θ = 30°, 150° |
10. Solve: \( \frac{\sqrt{3}}{\sin 2\alpha} + \frac{1}{\cos 2\alpha} = 4 \) \( \left[ 0^{\circ} \le \alpha \le 90^{\circ} \right] \)
Solution: \begin{align} \ & \frac{\sqrt{3}}{\sin 2\alpha} + \frac{1}{\cos 2\alpha} = 4\\ \text{or, } & \frac{ \sqrt{3} \cos 2\alpha + \sin 2\alpha }{\sin 2\alpha \cos 2\alpha } = 4 \\ \text{or, } & \sqrt{3} \cos 2\alpha + \sin 2\alpha = 4 \sin 2\alpha \cos 2\alpha \\ \text{or, } & \sqrt{3} \cos 2\alpha + \sin 2\alpha = 2 ( 2 \sin 2\alpha \cos 2\alpha ) \\ \text{or, } & \sqrt{3} \cos 2\alpha + \sin 2\alpha = 2\sin 4\alpha \\ \text{or, } & \sin 2\alpha +\sqrt{3} \cos 2\alpha = 2\sin 4\alpha ......(i) \end{align} \begin{align} \ \text{Now} & \\ \ & \sqrt{(\text{coeff. of } \sin 2\alpha )^2 + (\text{ coeff. of } \cos 2\alpha )^2 } \\ \ & = \sqrt{1^2+ ( \sqrt{3} )^2 }\\ & = \sqrt{1+3}\\ & = \sqrt{4}\\ & = 2 \end{align} \begin{align} \ & \text{ Dividing both sides of } eq^n(i) \text{ by } 2\\ \ & \text{We get, }\\ \ & \frac{1}{2} \sin 2\alpha + \frac{\sqrt{3}}{2} \cos 2\alpha = \frac{2\sin 4\alpha }{2} \\ \text{or, } & \sin 2\alpha \times \frac{1}{2} + \cos 2\alpha \times \frac{\sqrt{3}}{2} = \sin 4\alpha \\ \text{or, } & \sin 2\alpha \cos 60^{\circ} + \cos 2\alpha \sin 60^{\circ} = \sin 4\alpha \\ \text{or, } & \sin ( 2\alpha + 60^{\circ} ) = \sin 4\alpha, \sin (180^{\circ} - 4\alpha), \sin (3\times 180^{\circ} - 4\alpha ) \\ \text{or, } & 2\alpha + 60^{\circ} = 4\alpha , 180^{\circ} - 4\alpha, 3\times 180^{\circ} - 4\alpha \\ \end{align}
| Either | Or | Or |
|---|---|---|
| 2α + 60° = 4α | 2α + 60° = 180° - 4α | 2α + 60° = 3 × 180° - 4α |
| or, 60° = 4α - 2α | or, 2α + 4α = 180° - 60° | or, 2α + 4α = 540° - 60° |
| or, 60° = 2α | or, 6α = 120° | or, 6α = 480° |
| or, α = 60° / 2 | or, α = 120° / 6 | or, α = 480° / 6 |
| or, α = 30° | or, α = 20° | or, α = 80° |
11. (a) If \( 2\sin x \sin y = \frac{\sqrt{3}}{2}, \ \ \ \frac{1}{\tan x} + \frac{1}{\tan y } = 2 \) , find the minimum value of \( x + y \) .
Solution: \begin{align} \text{Given} \ & \\ \ & 2\sin x \sin y = \frac{\sqrt{3}}{2} ......(i) \\ \text{Now} \ & \\ \ & \frac{1}{\tan x} + \frac{1}{\tan y } = 2 \\ \text{or, } & \cot x + \cot y = 2 \\ \text{or, } & \frac{\cos x}{\sin x} + \frac{ \cos y }{\sin y} = 2 \\ \text{or, } & \frac{ \cos x \sin y +\sin x \cos y }{ \sin x \sin y } = 2 \\ \text{or, } & \sin x \cos y + \cos x \sin y = 2 \sin x \sin y \\ \text{or, } & \sin (x+y) = \frac{\sqrt{3}}{2}\\ \text{or, } & \sin (x+y) = \sin 60^{\circ} \\ \therefore & \ \ x+y = 60^{\circ} \end{align}
(b) \( 2\cos x \sin y = \frac{1}{\sqrt{2}} \ \ ( x> y ), \) \( \tan x + \cot y = 2 \) , find \( x-y \) . \( [ 0^{\circ} \le (x-y) \le 360^{\circ} ] \)
Solution: \begin{align} \text{Given} \ & \\ \ & 2\cos x \sin y = \frac{1}{\sqrt{2}} \ \ ( x> y ), ......(i) \\ \text{Now} \ & \\ \ & \tan x + \cot y = 2 \\ \text{or, } & \frac{\sin x}{\cos x} + \frac{ \cos y }{\sin y} = 2 \\ \text{or, } & \frac{ \sin x \sin y +\cos x \cos y }{ \cos x \sin y } = 2 \\ \text{or, } & \cos x \cos y + \sin x \sin y = 2 \cos x \sin y \\ \text{or, } & \cos (x-y) = \frac{1}{\sqrt{2}}\\ \text{or, } & \cos (x-y) = \cos 45^{\circ} , \cos (360^{\circ} - 45^{\circ}) \\ \therefore & \ \ x- y = 45^{\circ}, 315^{\circ} \end{align}
6 Comments
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