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Trigonometric Equation - JEE Advanced Previous Year Questions with Solutions

Trigonometric Equations is an important JEE Mathematics chapter that focuses on solving complex trigonometric identities and equations using transformations, inverse trigonometric concepts, and advanced problem-solving techniques, supported by previous-year JEE questions.

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Trigonometric Equation - JEE Advanced Previous Year Questions with Solutions

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Q. 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 [JEE 2010, 3]
Ans. 3
Q. 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 [JEE 2011, 4]
Ans. 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
Q. 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) $\frac{3 \pi}{2}<\varphi<2 \pi$ [JEE 2012, 4M]
Ans. (A,C,D)
Q. 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 [JEE(Advanced)-2014, 3(–1)]
Ans. (D)
Q. 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 [JEE 2015, 4M, –0M]
Ans. 8
Q. 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) $\frac{5 \pi}{9}$ [JEE(Advanced)-2016]
Ans. (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 $(-\pi, \pi)$ $\Rightarrow \operatorname{sum}=0$
Q. 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 $-$ $=$ [JEE(Advanced)-2018]
Ans. 0.5

Frequently Asked Questions

Find answers to common questions.

How many questions from Trigonometric Equations appear in JEE Advanced each year?

JEE Advanced typically sets 0–1 question directly from Trigonometric Equations per year. Between 2010 and 2026, seven questions appeared across eight years. The topic is rarely isolated — it usually combines with trigonometric identities or inequalities. Consistent practice of PYQs is the most reliable way to prepare, as per NTA's official syllabus guidelines.


Is Trigonometric Equations important for JEE Advanced or only JEE Main?

Trigonometric Equations is important for both, but JEE Advanced tests it at a higher difficulty level. JEE Main questions typically ask for the general solution. JEE Advanced questions ask for the number of solutions in a restricted domain, the sum of solutions, or combine it with additional algebraic constraints — requiring deeper conceptual clarity.

What is the general solution formula for sinθ = sinα?

The general solution of $\sin\theta = \sin\alpha$ is $\theta = n\pi + (-1)^n\alpha$, where $n \in \mathbb{Z}$. For $\cos\theta = \cos\alpha$, it is $\theta = 2n\pi \pm \alpha$. For $\tan\theta = \tan\alpha$, it is $\theta = n\pi + \alpha$. Memorising and applying these three forms correctly is the foundation for every JEE Advanced question in this chapter.

Why did the 2014 question (sinx + 2sin2x − sin3x = 3) have no solution?

The equation has no solution because the left-hand side cannot equal 3 for any $x \in (0, \pi)$. While individual terms can each reach their maximum values (sinx=1, 2sin2x=2, −sin3x=1), these maxima occur at different values of x that cannot coincide simultaneously. The actual maximum of the combined expression in this domain is less than 3.

What is the difficulty level of Trigonometric Equations in JEE Advanced compared to JEE Main?

JEE Advanced questions are significantly harder. JEE Main asks straightforward general solution problems (difficulty: 3/10). JEE Advanced combines domain restrictions, simultaneous conditions, and identity manipulation (difficulty: 7–9/10). The 2012 question (Q3 above), which had three simultaneous constraints, is rated among the top-10 hardest maths questions from that paper by eSaral's IIT Bombay faculty.


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Comments

Ankit Bharti
Aug. 17, 2024, 6:35 a.m.
Best pyq question 🤟🫣for jee aspirants.
Abhinit
May 26, 2024, 3:27 p.m.
Tankyou e Saral for ur help
Abhinit
May 26, 2024, 6:35 a.m.
Tankyou e Saral for ur help
Mridul Agarwaal
Oct. 15, 2022, 2:55 p.m.
Good Questions!
Gaurav
July 26, 2023, 6:35 a.m.
Hi
cluigik_889
Dec. 25, 2024, 6:35 a.m.
gay
Pratyaksh Goswami
July 19, 2025, 6:35 a.m.
bhaiya are you air 1
Siya
Jan. 16, 2022, 1:36 p.m.
Good question 👍👍 They can also be solve by different methods.🤔
Arjun pratap Singh
July 3, 2021, 6:42 p.m.
Op
Shibz
April 9, 2021, 9:38 p.m.
Awesome...but can be solved in da methods too
Mihir bessanio ( shibi bessanios daddy)
March 5, 2021, 5:03 p.m.
Good beta. Keep studying
Shibi bessanio
Feb. 18, 2021, 8:39 p.m.
Nice question but those can be solved by further methods so not harder
Shruti Wadatkar
Nov. 28, 2020, 9:07 a.m.
Great ! Thanks for it .
JAMES S MURRAY
Nov. 5, 2020, 8:48 p.m.
LIKE A BAWWS!!!
Reshma
Sept. 24, 2020, 9:07 a.m.
Problems are tough but thanks for providing these questions for practise🙏🙏🙏
Harshmohan
Aug. 11, 2020, 10:56 a.m.
Nice questions thank you
Israel
June 27, 2020, 11:20 a.m.
👌❣️👌
boss
April 20, 2020, 11:33 a.m.
good
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