Chapter 3

Solve the following trigonometric equations: Find the number of solution of the equation \(\sin 5 x \cdot \cos 3 x=\sin 6 x \cdot \cos 2 x\) in \([0, \pi]\).

Solve the following equations and tick the correct one. The number of solution of the equation \(\tan x \cdot \tan 4 x=1\), \(0

Solve: \(4 \sin ^{4} x+\cos ^{4} x=1\)

Solve the following trigonometric equations: Find the number of solution of the equation \(\cos 3 x \cdot \tan 5 x=\sin 7 x\) lying in \(\left[0, \frac{\pi}{2}\right]\).

Solve the following equations and tick the correct one. The number of solution of the equation \(12 \cos ^{3} x-7 \cos ^{2} x+4 \cos x-9=0\), is (a) 0 (b) 2 (c) infinity (d) None

Solve: \(4 \cos ^{2} x \sin x-2 \sin ^{2} x=2 \sin x\)

Solve the following trigonometric equations: The angles \(B\) and \(C(B>C)\) of a triangle satisfying the equation \(2 \tan x-\lambda\left(1+\tan ^{2} x\right)=0\), then find the angle \(A\), if \(0<\lambda<1\).

Solve the following equations and tick the correct one. The sum of all solution of the equation \(\cos \theta \cdot \cos \left(\frac{\pi}{3}+\theta\right) \cdot \cos \left(\frac{\pi}{3}-\theta\right)=\frac{1}{4}\) is (a) \(15 \pi\) (b) \(30 \pi\) (c) \(\frac{100 \pi}{3}\) (d) None

Solve: \(\sin ^{6} x+\cos ^{6} x=\frac{7}{16}\)

Solve the following trigonometric equations: Determine all values of ' \(a\) ' for which the equation \(\cos ^{4} x-(a+2) \cos ^{2} x-(a+3)=0\) has a solution and find those.

Solve the following equations and tick the correct one. The number of solution of \(16^{\sin ^{2} x}+16^{\cos ^{2} x}=10\), \(0 \leq x \leq 2 \pi\), is (a) 2 (b) 4 (c) 6 (d) 8 .

Solve: \(\sin 7 x+\sin 4 x+\sin x=0,0 \leq x \leq \frac{\pi}{2}\)

Solve the following trigonometric equations: Find all the solution of the equation \(\sin x+\sin \frac{\pi}{8}\left(\sqrt{(1-\cos x)^{2}+\sin ^{2} x}\right)=0\) in \(\left[\frac{5 \pi}{2}, \frac{7 \pi}{2}\right]\)

Solve the following equations and tick the correct one. The smallest positive value of \(x\) such that \(\tan \left(x+20^{\circ}\right)=\tan \left(x+10^{\circ}\right) \cdot \tan x \cdot \tan \left(x-10^{\circ}\right)\), is (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(60^{\circ}\) (d) \(75^{\circ}\)

Solve: \(\cos 3 x+\cos 2 x\) \(=\sin \left(\frac{3 x}{2}\right)+\sin \left(\frac{x}{2}\right), 0 \leq x \leq 2 \pi\)

Solve the following trigonometric equations: If the equation \(\sin ^{4} x-(k+2) \sin ^{2} x-(k+3)=0\) has a solution, then find the value of \(k\).

Solve the following equations and tick the correct one. The maximum value of \(\sin \left(x+\frac{\pi}{6}\right)+\cos \left(x+\frac{\pi}{6}\right)\) in \(\left(0, \frac{\pi}{2}\right)\) is attained at (a) \(\frac{\pi}{12}\) (b) \(\frac{\pi}{6}\) (c) \(\frac{\pi}{3}\) (d) \(\frac{\pi}{2}\)

Solve the following trigonometric equations: Find the number of principal solutions of the equation \(4.16^{\sin ^{2} x}=2^{6 \sin x} .\)

Solve the following equations and tick the correct one. The minimum value of \(2^{\sin x}+2^{\cos x}\) is (a) 1 (b) \(2^{1-\frac{1}{\sqrt{2}}}\) (c) \(2^{-\frac{1}{\sqrt{2}}}\) (d) \(\left(2-\frac{1}{\sqrt{2}}\right)\)

Solve: \(\cos 2 x+\cos 4 x=2 \cos x\)

Solve the following equations and tick the correct one. If \(\cos p \theta+\cos q \theta=0\), then the different values of \(\theta\) are in A.P., whose common difference is (a) \(\frac{\pi}{p+q}\) (b) \(\frac{\pi}{p-q}\) (c) \(\frac{2 \pi}{p \pm q}\) (d) \(\frac{3 \pi}{p \pm q}\)

Solve: \(\sin 2 x+\sin x+\cos 2 x+\cos x+1=0\)

Solve the following equations and tick the correct one. If \(\tan 2 x \cdot \tan x=1\), then \(x\) is (a) \(\frac{\pi}{3}\) (b) \((6 n \pm 1) \frac{\pi}{6}\) (c) \((4 n \pm 1) \frac{\pi}{6}\) (d) \((2 n \pm 1) \frac{\pi}{6}\)

Solve: \(\cos x \cos 2 x \cos 4 x=\frac{1}{4}\) \(0 \leq x \leq \pi\)

Solve the following equations and tick the correct one. The maximum value of \(5 \sin \theta+3 \sin (\theta-\alpha)\) is 7 , then the set of all possible values of \(\alpha\) is (a) \(\left(2 n \pi \pm \frac{\pi}{3}\right)\) (b) \(\left(2 n \pi \pm \frac{2 \pi}{3}\right)\) (c) \(\left[\frac{\pi}{3}, \frac{2 \pi}{3}\right]\) (d) None

Solve: \(\sin 3 \alpha=4 \sin \alpha \cdot \sin (x+\alpha) \cdot \sin (x-\alpha)\)

Solve the following equations and tick the correct one. If \(\tan \left(\frac{\pi}{2} \sin \theta\right)=\cot \left(\frac{\pi}{2} \sin \theta\right)\), then \(\sin \theta+\cos \theta\) is (a) \(2 n-1\) (b) \(2 n+1\) (c) \(2 n\) (d) \(n\)

Solve: \(\sin 2 x \sin 4 x+\cos 2 x=\cos 6 x\)

Solve the following equations and tick the correct one. If \(\sin \left(\frac{\pi}{4} \cot \theta\right)=\cos \left(\frac{\pi}{4} \tan \theta\right)\), then \(\theta\) is (a) \(\left(n \pi+\frac{\pi}{4}\right)\) (b) \(\left(2 n \pi \pm \frac{\pi}{4}\right)\) (c) \(\left(n \pi-\frac{\pi}{4}\right)\) (d) \(\left(2 n \pi \pm \frac{\pi}{6}\right)\)

Solve: \(\sin 3 x \cdot \cos x+\sin ^{2} x \cos ^{2} x\) \(+\sin x \cdot \cos ^{3} x=1,0 \leq x \leq 2 \pi\)

Solve the following equations and tick the correct one. If \(\tan (\pi \cos \theta)=\cot (\pi \sin \theta)\), then the values of \(\cos \left(\theta-\frac{\pi}{4}\right)\) is (are) (a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt{2}}\) (c) \(\pm \frac{1}{2 \sqrt{2}}\) (d) None

If \(\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}\) be the four roots of the equation \(\sin (\theta+\alpha)=k \sin 2 \theta\), no two of which differ by a multiple of \(2 \pi\), then prove that \(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=(2 n+1) \frac{\pi}{4}, n \in Z\)

Solve the following equations and tick the correct one. If \(3 \tan \left(\theta-15^{\circ}\right)=\tan \left(\theta+15^{\circ}\right)\), then \(\theta\) is (a) \(\left(n \pi+\frac{\pi}{4}\right)\) (b) \(\left(n \pi+\frac{\pi}{8}\right)\) (c) \(\left(n \pi+\frac{\pi}{3}\right)\) (d) None

Solve the following equations and tick the correct one. If \(\tan \theta+\tan \left(\theta+\frac{\pi}{3}\right)+\tan \left(\theta+\frac{2 \pi}{3}\right)=3\), then \(\theta\) is (a) \((2 n+1) \frac{\pi}{12}\) (b) \(\left(n \pi \pm \frac{\pi}{3}\right)\) (c) \((4 n+1) \frac{\pi}{12}\) (d) None

Solve the following equations and tick the correct one. The equation a \(\sin 2 x+\cos 2 x=2 a-7\) posses a solution if (a) \(a>6\) (b) \(2 \leq a \leq 6\) (c) \(a>2\) (d) None

Solve the following equations and tick the correct one. If \(a_{1}+a_{2} \sin x+a_{3} \cos x+a_{4} \sin 2 x+a_{5} \cos 2 x=0\) holds for all \(x\), then the number of possible 5 -tuplets is (a) 0 (b) 1 (c) 2 (d) infinity

Solve the following equations and tick the correct one. The number of solution of the equation \(1+\sin x \cdot \sin ^{2} \frac{x}{2}=0\) in \([-\pi, \pi]\) is (a) 0 (b) 1 (c) 2 (d) 3

Solve the following equations and tick the correct one. The solution of \(\sin ^{4} x+\cos ^{4} x+\sin 2 x+\alpha=0\) is solvable for (a) \(-\frac{1}{2} \leq \alpha \leq \frac{1}{2}\) (b) \(-3 \leq \alpha \leq 1\) (c) \(-\frac{3}{2} \leq \alpha \leq \frac{1}{2}\) (d) \(-1 \leq \alpha \leq 1\)

Solve the following equations and tick the correct one. The equation \(\sin ^{4} x-2 \cos ^{2} x+a^{2}=0\) is solvable for (a) \(-\sqrt{3} \leq a \leq \sqrt{3}\) (b) \(-\sqrt{2} \leq a \leq \sqrt{2}\) (c) \(-1 \leq a \leq 1\) (d) None

Solve the following equations and tick the correct one. The number of pairs \((x, y)\) satisfying the equations \(\sin x+\sin y=\sin (x+y)\) and \(|x|+|y|=1\), is (a) 2 (b) 4 (c) 6 (d) infinity

Solve the following equations and tick the correct one. The value of ' \(a\) ' for which the equation \(4 \operatorname{cosec}^{2}[\pi(a+x)]+a^{2}-4 a=0\), has a real solution, if (a) \(a=1\) (b) \(a=2\) (c) \(a=3\) (d) None

Solve the following equations and tick the correct one. If \(\sin x+\cos x=\sqrt{y+\frac{1}{y}}, x \in[0, \pi]\), then (a) \(x=\frac{\pi}{4}, y=1\) (b) \(y=0\) (c) \(y=2\) (d) \(x=\frac{3 \pi}{4}\)

Solve the following equations and tick the correct one. \(|\tan x+\sec x|=|\tan x|+|\sec x|, x \in[0,2 \pi]\), if \(x\) belongs to that interval (a) \([0, \pi]\) (b) \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) (c) \(\left[0, \frac{3 \pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right]\) (d) \((\pi, 2 \pi]\)

Solve the following equations and tick the correct one. The number of solutions of \(\sum^{5} \cos (r x)=5\) in the interval \([0,2 \pi]\) is (a) 0 (b) 1 (c) 5 (d) 10

Solve the following equations and tick the correct one. If \(f(x)=\max \\{\tan x, \cot x\\} .\) The number of roots of the cquation \(f(x)=\frac{1}{2+\sqrt{3}}\) in \((0,2 \pi)\) is (a) 0 (b) 2 (c) 4 (d) \(\infty\)

Solve the following equations and tick the correct one. The equation \(\cos 4 x-(\lambda+2) \cos 2 x-(\lambda+3)=0\) possesses a solution if (a) \(\lambda>-3\) (b) \(\lambda<-2\) (c) \(-3<\lambda<-2\) (d) \(\lambda \in z^{+}\)

Solve the following equations and tick the correct one. If \(0<\theta<2 \pi\) and \(2 \sin ^{2} \theta-5 \sin \theta+2>0\), then the range of \(\theta\) is (a) \(\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)\) (b) \(\left(0, \frac{5 \pi}{6}\right) \cup(\pi, 2 \pi)\) (c) \(\left(0, \frac{\pi}{6}\right) \cup(\pi, 2 \pi)\) (d) None.

Solve the following equations and tick the correct one. The number of values of \(x\) for which \(\sin 2 x+\cos 4 x=2\) is (a) 0 (b) \(\mathbb{l}\) (c) 2 (d) \(\infty\)

Solve the following equations and tick the correct one. The number of solutions of the equation \(x^{3}+x^{2}+4 x+2 \sin x=0\) in \(0

Solve the following equations and tick the correct one. The number of solutions of the equation \(\tan x+\sec x=2 \cos x\) lying in the interval \([0,2 \pi]\) is (a) 0 (b) 1 (c) 2 (d) 3