Trigonometric Equation - JEE Advanced Previous Year Questions with Solutions

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 Download eSaral app for free study material and video tutorials. [esquestion] The number of values of $\theta$ in the interval $\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$ such that $\theta \neq \frac{n \pi}{5}$ for $n=0, \pm 1, \pm 2$ and $\tan \theta=\cot 5 \theta$ as well as $\sin 2 \theta=\cos 4 \theta,$ is #tag# [JEE 2010, 3] #sol# 3 [/esquestion] [esquestion] The positive integer value of n > 3 satisfying the equation $\frac{1}{\sin \left(\frac{\pi}{\mathrm{n}}\right)}=\frac{1}{\sin \left(\frac{2 \pi}{\mathrm{n}}\right)}+\frac{1}{\sin \left(\frac{3 \pi}{\mathrm{n}}\right)}$ is #tag# [JEE 2011, 4] #sol# 7 $\frac{1}{\sin \frac{\pi}{\mathrm{n}}}=\frac{1}{\sin \frac{2 \pi}{\mathrm{n}}}+\frac{1}{\sin \frac{3 \pi}{\mathrm{n}}}$ $\Rightarrow \frac{1}{\sin \frac{\pi}{\mathrm{n}}}-\frac{1}{\sin \frac{3 \pi}{\mathrm{n}}}=\frac{1}{\sin \frac{2 \pi}{\mathrm{n}}}$ $\Rightarrow \frac{\sin \frac{3 \pi}{n}-\sin \frac{\pi}{n}}{\sin \frac{\pi}{n} \sin \frac{3 \pi}{n}}=\frac{1}{\sin \frac{2 \pi}{n}}$ $\Rightarrow \frac{2 \cos \frac{2 \pi}{\mathrm{n}} \sin \frac{\pi}{\mathrm{n}}}{\sin \frac{\pi}{\mathrm{n}} \sin \frac{3 \pi}{\mathrm{n}}}=\frac{1}{\sin \frac{2 \pi}{\mathrm{n}}}$ $\Rightarrow 2 \cos \frac{2 \pi}{\mathrm{n}} \sin \frac{2 \pi}{\mathrm{n}}=\sin \frac{3 \pi}{\mathrm{n}}$ $\Rightarrow \sin \frac{4 \pi}{\mathrm{n}}=\sin \frac{3 \pi}{\mathrm{n}} \Rightarrow \frac{4 \pi}{\mathrm{n}}=\mathrm{K} \pi+(-1)^{\mathrm{K}} \frac{3 \pi}{\mathrm{n}}$ If $\mathrm{K}=2 \mathrm{m} \quad \Rightarrow \quad \frac{\pi}{\mathrm{n}}=2 \mathrm{m} \pi$ $\Rightarrow \quad n=\frac{1}{2 m} \quad \Rightarrow n=\frac{1}{2}, \frac{1}{4}, \frac{1}{6} \ldots \ldots$ If $\mathrm{K}=2 \mathrm{m}+1 \Rightarrow \frac{7 \pi}{\mathrm{n}}=(2 \mathrm{m}+1) \pi$ $\Rightarrow \mathrm{n}=\frac{7}{2 \mathrm{m}+1} \quad \Rightarrow \quad \mathrm{n}=7, \frac{7}{3}, \frac{7}{5} \ldots \ldots$ Possible value of n is 7 [/esquestion] [esquestion] Let $\theta, \varphi \in[0,2 \pi]$ be such that $2 \cos \theta(1-\sin \varphi)=\sin ^{2} \theta\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right) \cos \varphi-1, \tan (2 \pi-\theta)>0$ and $-1<\sin \theta<-\frac{\sqrt{3}}{2} .$ Then $\varphi$ cannot satisfy- (A) $0<\varphi<\frac{\pi}{2}$ (B) $\frac{\pi}{2}<\varphi<\frac{4 \pi}{3}$ (C) $\frac{4 \pi}{3}<\varphi<\frac{3 \pi}{2}$ (D) $\frac{3 \pi}{2}<\varphi<2 \pi$ #tag# [JEE 2012, 4M] #sol# (A,C,D) [/esquestion] [esquestion] For $\mathrm{x} \in(0, \pi),$ the equation $\sin \mathrm{x}+2 \sin 2 \mathrm{x}-\sin 3 \mathrm{x}=3 \mathrm{has}$ (A) infinitely many solutions (B) three solutions (C) one solution (D) no solution #tag# [JEE(Advanced)-2014, 3(–1)] #sol# (D) [/esquestion] [esquestion] The number of distinct solutions of equation $\frac{5}{4} \cos ^{2} 2 x+\cos ^{4} x+\sin ^{4} x+\cos ^{6} x+\sin ^{6} x=2$ in the interval $[0,2 \pi]$ is #tag# [JEE 2015, 4M, –0M] #sol# 8 [/esquestion] [esquestion] Let $S=\left\{x \in(-\pi, \pi): x \neq 0, \pm \frac{\pi}{2}\right\} .$ The sum of all distinct solution of the equation $\sqrt{3} \sec x+\csc x+2(\tan x-\cot x)=0$ in the set $S$ is equal to $-$ (A) $-\frac{7 \pi}{9}$ (B) $-\frac{2 \pi}{9}$ (C) 0 (D) $\frac{5 \pi}{9}$ #tag# [JEE(Advanced)-2016] #sol# (C) $\sqrt{3} \sin x+\cos x=2 \cos 2 x$ $\Rightarrow \cos 2 x=\cos \left(x-\frac{\pi}{3}\right)$ $\Rightarrow 2 x=2 n \pi \pm\left(x-\frac{\pi}{3}\right)$ $\quad \quad x=(6 n-1) \frac{\pi}{3}$ or $(6 n+1) \frac{\pi}{9}$ $\Rightarrow x=-\frac{\pi}{3}, \frac{\pi}{9}, \frac{7 \pi}{9}$ and $-\frac{5 \pi}{9}$ in $(-\pi, \pi)$ $\Rightarrow \operatorname{sum}=0$ [/esquestion] [esquestion] Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3} a \cos x+2 b \sin x=c, \quad x$ $\in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha+\beta=\frac{\pi}{3} .$ Then the value of $\frac{b}{a}$ is $-$ $=$ #tag# [JEE(Advanced)-2018] #sol# 0.5 [/esquestion]

These questions test identity recall under pressure — bookmark the trigonometry formula page rather than looking things up mid-problem.

After completing Trigonometric Equations - JEE Advanced PYQs (Maths), use the JEE Advanced exam paper to review the wider question mix.