Trigonometry

22 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q18·MCQ·Easy–moderate
Open
Compound angle formulae
The value of $\operatorname{cosec}10^\circ - \sqrt3\,\sec10^\circ$ is equal to:
1. (1)4
2. (2)2
3. (3)8
4. (4)6
MathsJEE Main 22 Jan 2026, Shift 1·Q24·Numerical·Moderate
Open
Product formulae and standard angles
If $\dfrac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \dfrac{\alpha + \beta\sqrt5}{2}$ , where $\alpha, \beta \in \mathbb{N}$ , then $\alpha + \beta$ is equal to .
MathsJEE Main 22 Jan 2026, Shift 2·Q23·Numerical·Moderate
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Compound angle formulae
Let $\cos(\alpha + \beta) = -\dfrac{1}{10}$ and $\sin(\alpha - \beta) = \dfrac38$ , where $0 < \alpha < \dfrac{\pi}{3}$ and $0 < \beta < \dfrac{\pi}{4}$ . If $\tan2\alpha = \dfrac{3\left(1 - r\sqrt5\right)}{\sqrt{11}\left(s + \sqrt5\right)}$ , $r, s \in \mathbb{N}$ , then $r + s$ is equal to .
MathsJEE Main 23 Jan 2026, Shift 1·Q16·MCQ·Easy–moderate
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Trigonometric equations in an interval
Number of solutions of $\sqrt3\cos2\theta + 8\cos\theta + 3\sqrt3 = 0$ , $\theta \in [-3\pi, 2\pi]$ is:
1. (1)0
2. (2)5
3. (3)3
4. (4)4
MathsJEE Main 23 Jan 2026, Shift 1·Q18·MCQ·Moderate
Open
Maximum and minimum of trigonometric expressions
Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right),\quad \theta \in \mathbb{R}.$ Then $\alpha + 2\beta$ is equal to:
1. (1)4
2. (2)5
3. (3)3
4. (4)6
MathsJEE Main 23 Jan 2026, Shift 2·Q2·MCQ·Moderate
Open
Compound angle formulae
Let $\dfrac{\pi}{2} < \theta < \pi$ and $\cot\theta = -\dfrac{1}{2\sqrt2}$ . Then the value of $\sin\left(\dfrac{15\theta}{2}\right)(\cos8\theta + \sin8\theta) + \cos\left(\dfrac{15\theta}{2}\right)(\cos8\theta - \sin8\theta)$ is equal to:
1. (1) $\dfrac{1-\sqrt2}{\sqrt3}$
2. (2) $-\dfrac{\sqrt2}{\sqrt3}$
3. (3) $\dfrac{\sqrt2-1}{\sqrt3}$
4. (4) $\dfrac{\sqrt2}{\sqrt3}$
We corrected a printing error in the original paper
Source printed “ $(\cos8 - \sin8\theta)$ ” (the $\theta$ of $\cos8\theta$ missing).
MathsJEE Main 23 Jan 2026, Shift 2·Q13·MCQ·Easy–moderate
Open
Range of trigonometric expressions
The least value of $(\cos^2\theta - 6\sin\theta\cos\theta + 3\sin^2\theta + 2)$ is
1. (1) $-1$
2. (2) $4 + \sqrt{10}$
3. (3) $4 - \sqrt{10}$
4. (4)1
MathsJEE Main 24 Jan 2026, Shift 1·Q8·MCQ·Easy–moderate
Open
Trigonometric identities and products
The value of $\dfrac{\sqrt3\operatorname{cosec}20^\circ - \sec20^\circ}{\cos20^\circ\cos40^\circ\cos60^\circ\cos80^\circ}$ is equal to
1. (1)32
2. (2)16
3. (3)64
4. (4)12
MathsJEE Main 24 Jan 2026, Shift 1·Q10·MCQ·Moderate
Open
Half angle and compound angles
If $\cot x = \dfrac{5}{12}$ for some $x \in \left(\pi, \dfrac{3\pi}{2}\right)$ , then $\sin7x\left(\cos\dfrac{13x}{2} + \sin\dfrac{13x}{2}\right) + \cos7x\left(\cos\dfrac{13x}{2} - \sin\dfrac{13x}{2}\right)$ is equal to
1. (1) $\dfrac{4}{\sqrt{26}}$
2. (2) $\dfrac{6}{\sqrt{26}}$
3. (3) $\dfrac{1}{\sqrt{13}}$
4. (4) $\dfrac{5}{\sqrt{13}}$
MathsJEE Main 24 Jan 2026, Shift 2·Q25·Numerical·Tough
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Trigonometric equations
The number of elements in the set $\left\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ)\tan x\tan(x - 50^\circ)\right\}$ is .
MathsJEE Main 28 Jan 2026, Shift 1·Q4·MCQ·Moderate
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Compound angle identities
If $\dfrac{\tan(A - B)}{\tan A} + \dfrac{\sin^2 C}{\sin^2 A} = 1$ , $A, B, C \in \left(0, \dfrac{\pi}{2}\right)$ , then
1. (1) $\tan A$ , $\tan C$ , $\tan B$ are in G.P.
2. (2) $\tan A$ , $\tan B$ , $\tan C$ are in G.P.
3. (3) $\tan A$ , $\tan C$ , $\tan B$ are in A.P.
4. (4) $\tan A$ , $\tan B$ , $\tan C$ are in A.P.
MathsJEE Main 28 Jan 2026, Shift 2·Q15·MCQ·Easy–moderate
Open
Signs of trigonometric functions
The sum of all the elements in the range of $f(x) = \operatorname{Sgn}(\sin x) + \operatorname{Sgn}(\cos x) + \operatorname{Sgn}(\tan x) + \operatorname{Sgn}(\cot x)$ , $x \neq \dfrac{n\pi}{2}$ , $n \in \mathbb{Z}$ , where $\operatorname{Sgn}(t) = \begin{cases} 1, & \text{if } t > 0\\ -1 & \text{if } t < 0\end{cases}$ , is
1. (1)4
2. (2)2
3. (3) $-2$
4. (4)0
MathsJEE Main 2 Apr 2026, Shift 1·Q11·MCQ·Easy–moderate
Open
Product of sines
If $\sin\left(\dfrac{\pi}{18}\right)\sin\left(\dfrac{5\pi}{18}\right)\sin\left(\dfrac{7\pi}{18}\right) = K$ , then the value of $\sin\left(\dfrac{10K\pi}{3}\right)$ is:
1. (1) $\dfrac{\sqrt3 + 1}{2\sqrt2}$
2. (2) $\dfrac{\sqrt3 - 1}{\sqrt2}$
3. (3) $\dfrac{\sqrt3}{2}$
4. (4) $\dfrac12$
MathsJEE Main 2 Apr 2026, Shift 1·Q12·MCQ·Moderate
Open
Trigonometric equations in an interval
Let $S = \{x \in [-\pi, \pi] : \sin x\,(\sin x + \cos x) = a,\ a \in \mathbb{Z}\}$ . Then $n(S)$ is equal to:
1. (1)3
2. (2)6
3. (3)7
4. (4)9
MathsJEE Main 2 Apr 2026, Shift 2·Q13·MCQ·Moderate
Open
Trigonometric identities and range
Let $P = \{\theta \in [0, 4\pi] : \tan^2\theta \neq 1\}$ and $S = \{a \in \mathbb{Z} : 2(\cos^8\theta - \sin^8\theta)\sec2\theta = a^2,\ \theta \in P\}.$ Then $n(S)$ is:
1. (1)0
2. (2)1
3. (3)2
4. (4)3
MathsJEE Main 4 Apr 2026, Shift 1·Q23·Numerical·Moderate
Open
Telescoping trigonometric sums
If $A = \dfrac{\sin 3^\circ}{\cos 9^\circ} + \dfrac{\sin 9^\circ}{\cos 27^\circ} + \dfrac{\sin 27^\circ}{\cos 81^\circ}$ and $B = \tan 81^\circ - \tan 3^\circ$ , then $\dfrac{B}{A}$ is equal to .
MathsJEE Main 4 Apr 2026, Shift 1·Q24·Numerical·Tough
Open
Trigonometric sum with doubling angles
Let $\vec a_k = (\tan\theta_k)\hat i + \hat j$ and $\vec b_k = \hat i - (\cot\theta_k)\hat j$ , where $\theta_k = \dfrac{2^{k-1}\pi}{2^n + 1}$ , for some $n \in \mathbb{N}$ , $n > 5$ . Then the value of $\dfrac{\displaystyle\sum_{k=1}^{n}\lvert\vec a_k\rvert^2}{\displaystyle\sum_{k=1}^{n}\lvert\vec b_k\rvert^2}$ is .
We corrected a printing error in the original paper
Source printed “ $\vec b_k = \hat i - (\cos\theta_k)\hat j$ ”; corrected to the intended “ $(\cot\theta_k)\hat j$ ” (with $\cos$ the ratio depends on $n$ , e.g. 166.5 for $n = 6$ , and is never 3; with $\cot$ it is 3 for every $n$ , as the key and the source's solution use).
MathsJEE Main 5 Apr 2026, Shift 1·Q6·MCQ·Moderate
Open
Compound angle formulae
Let $\tan A$ , $\tan B$ , where $A, B \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ , be the roots of the quadratic equation $x^2 - 2x - 5 = 0$ . Then $20\sin^2\left(\dfrac{A + B}{2}\right)$ is equal to:
1. (1) $10 + \sqrt{10}$
2. (2) $10 - 2\sqrt{10}$
3. (3) $10 - 3\sqrt{10}$
4. (4) $10 - \sqrt{10}$
MathsJEE Main 5 Apr 2026, Shift 1·Q12·MCQ·Easy–moderate
Open
Range of trigonometric expression
The sum of all the integral values of $p$ such that the equation $3\sin^2x + 12\cos x - 3 = p$ , $x \in \mathbb{R}$ , has at least one solution, is:
1. (1) $-54$
2. (2) $-60$
3. (3) $-75$
4. (4) $-84$
MathsJEE Main 5 Apr 2026, Shift 2·Q23·Numerical·Moderate
Open
Trigonometric equations
If $S = \left\{\theta \in [-\pi, \pi] : \cos\theta\cos\dfrac{5\theta}{2} = \cos7\theta\cos\dfrac{7\theta}{2}\right\}$ , then $n(S)$ is equal to .
MathsJEE Main 6 Apr 2026, Shift 1·Q14·MCQ·Easy–moderate
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General solution of trigonometric equations
Let $S = \left\{\theta \in (-2\pi, 2\pi) : \cos\theta + 1 = \sqrt3\sin\theta\right\}$ . Then $\displaystyle\sum_{\theta\in S}\theta$ is equal to:
1. (1) $-\dfrac{2\pi}{3}$
2. (2) $-\dfrac{4\pi}{3}$
3. (3) $\dfrac{2\pi}{3}$
4. (4) $\dfrac{4\pi}{3}$
MathsJEE Main 6 Apr 2026, Shift 1·Q24·Numerical·Moderate
Open
Extreme values of trigonometric functions
For the functions $f(\theta) = \alpha\tan^2\theta + \beta\cot^2\theta$ , and $g(\theta) = \alpha\sin^2\theta + \beta\cos^2\theta$ , $\alpha > \beta > 0$ , let $\displaystyle\min_{0<\theta<\frac{\pi}{2}} f(\theta) = \max_{0<\theta<\pi} g(\theta)$ . If the first term of a G.P. is $\left(\dfrac{\alpha}{2\beta}\right)$ , its common ratio is $\left(\dfrac{2\beta}{\alpha}\right)$ and the sum of its first 10 terms is $\dfrac{m}{n}$ , $\gcd(m, n) = 1$ , then $m + n$ is equal to .