Trigonometric Equations — JEE Main PYQs

59 previous year questions on Trigonometric Equations (Mathematics), newest first. Open any question for its complete step-by-step solution.

Number of solutions of $3 cos 2 θ + 8 cos θ + 33 = 0, θ \in [- 3 π, 2 π]$ is:
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The number of elements in the set ${x \in [0, 18 0^{\circ}] : tan (x + 10 0^{\circ}) = tan (x + 5 0^{\circ}) tan x tan (x - 5 0^{\circ})}$ is
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Let $S = {x \in [- π, π] : sin x (sin x + cos x) = a$ , $a \in Z}$ . Then $n (S)$ is equal to :
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If $S = {θ \in [- π, π] : cos θ cos \frac{5 θ}{2} = cos 7 θ cos \frac{7 θ}{2}}$ , then $n (S)$ is equal to
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Let $S = {θ \in (- 2 π, 2 π) : cos θ + 1 = 3 sin θ}$ .

Then $\sum_{θ \in S} θ$ is equal to:
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The sum of all values of $θ \in [0, 2 π]$ satisfying $2 sin^{2} θ = cos 2 θ$ and $2 cos^{2} θ = 3 sin θ$ is
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Let
$A = {x \in (0, π) - {\frac{π}{2}} : lo g_{(2/ π)} ∣ sin x ∣ + lo g_{(2/ π)} ∣ cos x ∣ = 2}$
and
$B = {x \geq 0 : x (x - 4) - 3∣ x - 2∣ + 6 = 0}$ . Then $n (A \cup B)$ is equal to:
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If $θ \in [- \frac{7 π}{6}, \frac{4 π}{3}]$ , then the number of solutions of $3 cosec^{2} θ - 2 (3 - 1) cosec θ - 4 = 0$ , is equal to
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The number of solutions of equation $(4 - 3) sin x - 23 cos^{2} x = - \frac{4}{1 + 3}, x \in [- 2 π, \frac{5 π}{2}]$ is
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If $θ \in [- 2 π, 2 π]$ , then the number of solutions of $22 cos^{2} θ + (2 - 6) cos θ - 3 = 0$ , is equal to:
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The number of solutions of the equation $2 x + 3 tan x = π$ , $x \in [- 2 π, 2 π] - {\pm \frac{π}{2}, \pm \frac{3 π}{2}}$ is
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The number of solutions of the equation
$cos 2 θ cos \frac{θ}{2} + cos \frac{5 θ}{2} = 2 cos^{3} \frac{5 θ}{2}$ in $[- \frac{π}{2}, \frac{π}{2}]$ is :
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Suppose $θ \in [0, \frac{π}{4}]$ is a solution of $4 cos θ - 3 sin θ = 1$ . Then $cos θ$ is equal to:
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The number of solutions, of the equation $e^{s i n x} - 2 e^{- s i n x} = 2$ is
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If $2 sin^{3} x + sin 2 x cos x + 4 sin x - 4 = 0$ has exactly 3 solutions in the interval $[0, \frac{nπ}{2}]$ , $n \in N$ , then the roots of the equation $x^{2} + n x + (n - 3) = 0$ belong to:
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If $2 tan^{2} θ - 5 sec θ = 1$ has exactly 7 solutions in the interval $[0, \frac{n π}{2}]$ , for the least value of $n \in N$ then $\sum_{k = 1}^{n} \frac{k}{2 ^{k}}$ is equal to :
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Let the set of all $a \in R$ such that the equation $cos 2 x + a sin x = 2 a - 7$ has a solution be $[p, q]$ and $r = tan 9^{\circ} - tan 2 7^{\circ} - \frac{1}{c o t 6 3 ^{\circ}} + tan 8 1^{\circ}$ , then pqr is equal to ____.
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The number of solutions of the equation $4 sin^{2} x - 4 cos^{3} x + 9 - 4 cos x = 0; x \in [- 2 π, 2 π]$ is :
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If $α$ , $- \frac{π}{2} < α < \frac{π}{2}$ is the solution of $4 cos θ + 5 sin θ = 1$ , then the value of $tan α$ is
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Let $S = {x \in (- \frac{π}{2}, \frac{π}{2}) : 9^{1 - t a n^{2} x} + 9^{t a n^{2} x} = 10}$ and $β = \sum_{x \in s} tan^{2} (\frac{x}{3})$ , then $\frac{1}{6} (β - 14)^{2}$ is equa …
JEE Main 2023View solution →
The number of elements in the set $S = {θ \in [0, 2 π] : 3 cos^{4} θ - 5 cos^{2} θ - 2 sin^{6} θ + 2 = 0}$ is
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If the solution of the equation $lo g_{c o s x} cot x + 4 lo g_{s i n x} tan x = 1, x \in (0, \frac{π}{2})$ , is $sin^{- 1} (\frac{α + β}{2})$ , where $α, β$ are integers, then $α + β$ is equal to:
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If m and n respectively are the numbers of positive and negative value of $θ$ in the interval $[- π, π]$ that satisfy the equation $cos 2 θ cos \frac{θ}{2} = cos 3 θ cos \frac{9 θ}{2}$ , then mn is equal to ______.
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Let $S = {θ \in [0, 2 π) : tan (π cos θ) + tan (π sin θ) = 0}$ .

Then $\sum_{θ \in S} sin^{2} (θ + \frac{π}{4})$ is equal to
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The number of elements in the set $S = {x \in R : 2 cos (\frac{x ^{2} + x}{6}) = 4^{x} + 4^{- x}}$ is:
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Let $S = [- π, \frac{π}{2}) - {- \frac{π}{2}, - \frac{π}{4}, - \frac{3 π}{4}, \frac{π}{4}}$ . Then the number of elements in the set
$A = {θ \in S : tan θ (1 + 5 tan (2 θ)) = 5 - tan (2 θ)}$ is
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The number of solutions of $∣ cos x ∣ = sin x$ , such that $- 4 π \leq x \leq 4 π$ is:
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If the sum of solutions of the system of equations $2 sin^{2} θ - cos 2 θ = 0$ and $2 cos^{2} θ + 3 sin θ = 0$ in the interval $[0, 2 π]$ is $k π$ , then k is equal to ______.
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Let $S = {θ \in [0, 2 π] : 8^{2 s i n^{2} θ} + 8^{2 c o s^{2} θ} = 16}$ . Then $n (S) + \sum_{θ \in S} (sec (\frac{π}{4} + 2 θ) cosec (\frac{π}{4} + 2 θ))$ is equal to :
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The number of solutions of the equation $sin x = cos^{2} x$ in the interval $(0, 10)$ is ______.
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The number of elements in the set $S = {θ \in [- 4 π, 4 π] : 3 cos^{2} 2 θ + 6 cos 2 θ - 10 cos^{2} θ + 5 = 0}$ is ____.
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The number of solutions of the equation $2 θ - cos^{2} θ + 2 = 0$ in $R$ is equal to ____.
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Let $S = {θ \in (0, 2 π) : 7 cos^{2} θ - 3 sin^{2} θ - 2 cos^{2} 2 θ = 2}$ . Then, the sum of roots of all the equations $x^{2} - 2 (tan^{2} θ + cot^{2} θ) x + 6 sin^{2} θ = 0 θ \in S$ , is …
JEE Main 2022View solution →
Let
$S = {θ \in [- π, π] - {\pm \frac{π}{2}} : sin θ tan θ + tan θ = sin 2 θ}$ .
If $T = \sum_{θ \in S} cos 2 θ$ , then $T + n (S)$ is equal
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The number of solutions of the equation $cos (x + \frac{π}{3}) cos (\frac{π}{3} - x) = \frac{1}{4} cos^{2} 2 x, x \in [- 3 π, 3 π]$ is :
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The number of values of $x$ in the interval $(\frac{π}{4}, \frac{7 π}{4})$ for which $14 cosec^{2} x - 2 sin^{2} x = 21 - 4 cos^{2} x$ holds, is ____
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The sum of solutions of the equation $\frac{c o s x}{1 + s i n x} = ∣ tan 2 x ∣, x \in (- \frac{π}{2}, \frac{π}{2}) - {\frac{π}{4}, - \frac{π}{4}}$ is:
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Let $S$ be the sum of all solutions (in radians) of the equation $sin^{4} θ + cos^{4} θ - sin θ cos θ = 0$ in $[0, 4 π]$ . Then $\frac{8 S}{π}$ is equal to ______.
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All possible values of $θ \in [0, 2 π]$ for which $sin 2 θ + tan 2 θ > 0$ lie in :
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The number of solutions of $sin^{7} x + cos^{7} x = 1$ , $x \in [0, 4 π]$ is equal to:
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If $3 (cos^{2} x) = (3 - 1) cos x + 1$ , the number of solutions of the given equation when $x \in [0, \frac{π}{2}]$ is ____
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The number of solutions of the equation $3 2^{t a n^{2} x} + 3 2^{s e c^{2} x} = 81, 0 \leq x \leq \frac{π}{4}$ is:
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The sum of all values of $x$ in $[0, 2 π]$ , for which $sin x + sin 2 x + sin 3 x + sin 4 x = 0$ , is equal to :
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If $n$ is the number of solutions of the equation $2 cos x (4 sin (\frac{π}{4} + x) sin (\frac{π}{4} - x) - 1) = 1, x \in [0, π]$ and $S$ is the sum of all these solutions, then the order pair $(n, S)$ is:
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The number of roots of the equation,
$(81)^{s i n^{2} x} + (81)^{c o s^{2} x} = 30$ in the interval $[0, π]$ is equal to :
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The number of solution of the equation $x + 2 tan x = \frac{π}{2}$ in the interval $[0, 2 π]$ is:
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The number of solutions of the equations $∣ cot x ∣ = cot x + \frac{1}{s i n x}$ in the interval $[0, 2 π]$ is ____
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Let $α$ and $β$ be two real roots of the equation $(k + 1) tan^{2} x - 2 \cdot λ tan x = (1 - k)$ , where $k (\neq = - 1)$ and $λ$ are real numbers. If $tan^{2} (α + β) = 50$ , then a value of …
JEE Main 2020View solution →
If $θ_{1}$ and $θ_{2}$ be respectively the smallest and the largest values of $θ$ in $(0, 2 π) - {π}$ which satisfy the equation, $2 cot^{2} θ - \frac{5}{s i n θ} + 4 = 0$ , then $\int_{θ_{1}}^{θ_{2}} cos^{2} 3 θ d θ$ is equal to:
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The number of distinct solutions of the equation, $lo g_{\frac{1}{2}} ∣ sin x ∣ = 2 - lo g_{\frac{1}{2}} ∣ cos x ∣$ in the interval $[0, 2 π]$ , is ______.
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