Chapter 3

Solve the following trigonometric equations: \(\cot \left(\frac{x}{2}\right)-\operatorname{cosec}\left(\frac{x}{2}\right)=\cot x\)

Solve the following equations and tick the correct one. \(\sin ^{2} \theta-\cos \theta=\frac{1}{2}, 0 \leq \theta \leq 2 \pi\) (a) \(\frac{2 \pi}{3}, \frac{\pi}{3}\) (b) \(\frac{\pi}{3}, \frac{5 \pi}{3}\) (c) \(-\frac{\pi}{3}, \frac{2 \pi}{3}\) (d) \(\frac{2 \pi}{3}, \frac{5 \pi}{3}\)

Solve: \(7 \cos ^{2} \theta+3 \sin ^{2} \theta=4\)

Solve the following trigonometric equations: \(8 \cos x \cdot \cos 2 x \cdot \cos 4 x=\frac{\sin 6 x}{\sin x}\)

Solve the following equations and tick the correct one. If \(3 \tan ^{2} \theta-2 \sin \theta=0\), then \(\theta\) is (a) \(n \pi\) (b) \(n \pi+(-1)^{n} \frac{\pi}{6}\) (c) \(n \pi-(-1)^{n} \frac{\pi}{6}\) (d) \(n \pi+\frac{\pi}{3}\)

Solve: \(\sin 2 \theta+\sin 4 \theta+\sin 6 \theta=0\)

Solve the following trigonometric equations: \(\frac{\tan x}{\tan 2 x}+\frac{\tan 2 x}{\tan x}+2=0\)

Solve the following equations and tick the correct one. If \(\tan ^{2} x+(1-\sqrt{3}) \tan x-\sqrt{3}=0\), then \(x\) is (a) \(n \pi+\frac{\pi}{3}\) (b) \(n \pi-\frac{\pi}{3}\) (c) \(n \pi+\frac{\pi}{4}\) (d) \(n \pi-\frac{\pi}{4}\)

Solve: \(\tan ^{2} \theta+(1-\sqrt{3}) \tan \theta-\sqrt{3}=0\)

Solve the following trigonometric equations: Solve: \(\cos x \cos (6 x)=-1\)

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

Solve: \(\tan \theta+\tan 2 \theta+\tan \theta \tan 2 \theta=1\)

Solve the following trigonometric equations: Solve: \(\cos (4 x)+\sin (5 x)=2\)

Solve the following equations and tick the correct one. If \(\tan \theta+\cot \theta=2\), then \(\theta\) is (a) \(n \pi+\frac{\pi}{4}\) (b) \(n \pi-\frac{\pi}{4}\) (c) (d) \(n \pi-\frac{\pi}{3}\).

Solve: \(3 \tan \left(\theta-15^{\circ}\right)=\tan \left(\theta+15^{\circ}\right)\)

Solve the following trigonometric equations: \(\sin 2 x+5 \cos x+5 \sin x+1=0\)

Solve the following equations and tick the correct one. The set of values of \(x\) for which \(\frac{\tan 3 x-\tan 2 x}{1+\tan 3 x \tan 2 x}=1\) is (a) \(\phi\) (b) \(\frac{\pi}{4}\) (c) \(n \pi+\frac{\pi}{3}\) (d) \(2 n \pi+\frac{\pi}{4}\)

Solve: \(\tan \theta+\tan 2 \theta+\tan 3 \theta=0\)

Solve the following trigonometric equations: \(\sin x+\sin 2 x+\sin 3 x=\cos x+\cos 2 x+\cos 3 x\) in the interval \(0 \leq x \leq \pi\)

Solve the following equations and tick the correct one. If \(\sin 5 x+\sin 3 x+\sin x=0\), then the value of \(x\) other than zero, lying between \(0 \leq x \leq \frac{\pi}{2}\) (a) \(\frac{\pi}{6}\) (b) \(\frac{\pi}{12}\) (c) \(\frac{\pi}{3}\) (d) \(\frac{\pi}{4}\)

Solve: \(4 \sin \theta \sin 2 \theta \sin 4 \theta=\sin 3 \theta\)

Solve the following trigonometric equations: \(\sin ^{2} x \tan x+\cos ^{2} x \cot x-\sin 2 x\) \(=1+\tan x \cot x\)

Solve the following equations and tick the correct one. If \(\alpha\) and \(\beta\) are acute positive angles satisfying the equation \(3 \sin ^{2} \alpha+2 \sin ^{2} \beta=1\) and \(3 \sin 2 \alpha-2 \sin 2 \beta=0\), then \(\alpha+2 \beta\) is (a) 0 (b) \(\frac{\pi}{4}\) (c) \(\frac{\pi}{3}\) (d) \(\frac{\pi}{2}\)

Solve: \(\sqrt{2} \sec \theta+\tan \theta=1\)

Solve the following trigonometric equations: \(\sin ^{2} 4 x+\cos ^{2} x=2 \sin 4 x \cos ^{4} x\)

Solve the following equations and tick the correct one. If \(2 \sin ^{2} x+\sin ^{2} 2 x=2,-\pi

Solve the following trigonometric equations: \(\sin ^{4} x+\cos ^{4} x=\frac{7}{2} \sin x \cos x\)

Solve the following equations and tick the correct one. The real roots of the equation \(\cos ^{7} x+\sin ^{4} x=1\) in \((-\pi, \pi)\) are (a) \(-\frac{\pi}{2}, 0\) (b) \(-\frac{\pi}{2}, 0, \frac{\pi}{2}\) (c) \(0, \frac{\pi}{2}\) (d) \(0, \frac{\pi}{4}, \frac{\pi}{2}\)

Solve: \(\cos \theta+\sqrt{3} \sin \theta=2 \cos 2 \theta\)

Solve the following trigonometric equations: \(\sin ^{4} x+\cos ^{4} x\) \(=2 \cos \left(2 x+\frac{\pi}{6}\right) \cos \left(2 x-\frac{\pi}{6}\right)\)

Solve the following equations and tick the correct one. The general solution of \(\cos ^{5} x-\sin ^{5} x-1=0\) is (a) \(n \pi\) (b) \(2 n \pi\) (c) \(n \pi+\frac{\pi}{2}\) (d) \(2 n \pi+\frac{\pi}{2}\)

Solve: \(x+y=\frac{2 \pi}{3}\) and \(\cos x+\cos y=\frac{3}{2}\)

Solve the following trigonometric equations: \(\sin ^{4} x+\sin ^{4}\left(x+\frac{\pi}{4}\right)=\frac{1}{4}\)

If \(4 \sin ^{4} x+\cos ^{4} x=1\), then \(x\) is (a) \(n \pi\) (b) \(n \pi \pm \sin ^{-1} \sqrt{\frac{2}{5}}\) (c) \(\frac{2 n \pi}{3}\) (d) \(2 n \pi \pm \frac{\pi}{4}\)

Solve: \(x+y=\frac{\pi}{4}\) and \(\tan x+\tan y=1\)

Solve the following trigonometric equations: If \(\cos \left(x+\frac{\pi}{3}\right)+\cos x=a\), then find all values of a so that the equation has a real solution.

Solve the following equations and tick the correct one. The number of points of intersection of \(2 y=1\) and \(y=\cos x\) in \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) is (a) 1 (b) 2 (c) 3 (d) 4

Solve: \(r \sin \theta=3\) and \(r=4(1+\sin \theta), 0 \leq \theta \leq 2 \pi\)

Solve the following trigonometric equations: Find the number of roots of \(\cos x-x+\frac{1}{2}=0\) lies in \(\left(0, \frac{\pi}{2}\right)\)

Solve the following equations and tick the correct one. The number of values of \(x\) in the interval \([0,3 \pi]\) satisfying the equation \(2 \sin ^{2} x+5 \sin x-3=0\) is (a) 6 (b) 1 (c) 2 (d) 4

Solve: \(\sin x+\sin y=1, \cos 2 x-\cos 2 y=1\)

Solve the following trigonometric equations: Find the number of integral ordered pairs satisfy the equations \(\left\\{\begin{array}{l}\cos (x y)=x \\ \tan (x y)=y\end{array}\right.\)

Solve the following equations and tick the correct one. The number of values of \(x\) in \([0,5 \pi]\) satisfying the equation \(3 \sin ^{2} x-7 \sin x+2=0\) is (a) 0 (b) 5 (c) 6 (d) 10

If \(A\) and \(B\) are acute \(+\) ve angles satisfying the equations \(3 \sin ^{2} A+2 \sin ^{2} B=1\) and \(3 \sin 2 A=2 \sin 2 B\), then find \((A+2 B)\).

Solve the following trigonometric equations: Find the number of real solutions of \(\sin ^{2016} x-\cos ^{2016} x=1\) in \([0,2 \pi]\)

Solve the following equations and tick the correct one. The number of solution of the equation \(|\cot x|=\cot x+\frac{1}{\sin x}, 0

If \(\tan (A-B)=1\) and \(\sec (A+B)=\frac{2}{\sqrt{3}}\), then find the smallest +ve values of \(A\) and \(B\) and their most general values.

Solve the following trigonometric equations: Find the number of ordered pairs which satisfy the equation \(x^{2}+2 x \sin (x y)+1=0\) for \(y \in[0,2 \pi]\).

Solve the following equations and tick the correct one. The number of solution of \(|\cos x|=\sin x\) such that \(0

If \(\sin A=\sin B\) and \(\cos A=\cos B\), then find the values of \(A\) in terms of \(B\).