General value of ${\rm{\theta }}$ satisfying the equation ${\tan ^2}\,\,{\rm{\theta }} + \sec \,\,2{\rm{\theta }} = 1$ is

$m\pi ,\;\,n\pi + \frac{\pi }{3}$

$m\pi ,\;\,n\pi \;\,\, \pm \,\,\frac{\pi }{6}$

If $\sin \,\,{\rm{\theta }} = \sqrt 3 \,\,\cos \,\,{\rm{\theta }},{\rm{\;\pi }} < {\rm{\theta }} < 0$, then ${\rm{\theta }}$ is equal to

$ - \,\,\frac{{4{\rm{\pi }}}}{6}$

$ - \,\,\frac{{5{\rm{\pi }}}}{6}$

The solution of the equation $4{\sin ^4}x + {\cos ^4}x = 1$ is

$x = \left( {n + 2} \right)\pi $

If the solutions for ${\rm{\theta }},\,\cos p{\rm{\theta }} + \cos q{\rm{\theta }} = 0,\;\,p > 0,\;\,q > 0$ are in AP, then the numerically smallest common difference of AP is

$\frac{\pi }{{2\left( {p + q} \right)}}$

The general value of $x$ for which $\cos \,\,2x,\,\,\frac{1}{2}\;\,\,{\rm{and}}\,\,\sin \,\,2x$ are in AP, are given by

$n\pi ,\;\,n\pi + \frac{\pi }{2}$

$n\pi + \frac{\pi }{4},\,\;\frac{{3n\pi }}{4}$

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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 241-270 of 581 questions

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241 Easy

241 of 581

Analyzing +4 | -1

General value of

\mathrm{\theta }

satisfying the equation

{\tan ^2}\,\,\mathrm{\theta } + \sec \,\,2\mathrm{\theta } = 1

View answer and explanation

Correct option: B

m\pi ,\;\,n\pi \;\,\, \pm \,\,\frac{\pi }{3}

Option B (

m\pi ,\;\,n\pi \;\,\,\pm \,\,\frac{\pi }{3}

) is correct because it represents the general solutions obtained by solving the given trigonometric equation.

242 Easy

242 of 581

Application +4 | -1

\sin \,\,\left( {\pi \,\,\cot \,\,\theta } \right) = \cos \,\,\left( {\pi \,\,\tan \,\,\theta } \right)

then

\cot \,\,2\;\theta

is equal to where

n \in Z

View answer and explanation

Correct option: B

n + \frac{1}{4}

Option B (

n + \frac{1}{4}

) is correct because it is the simplified form of the expression after applying trigonometric identities and solving for

\cot 2\theta

243 Medium

243 of 581

Analyzing +4 | -1

The solution of trigonometric equation

\cos ^4 x+\sin ^4 x=2 \cos (2 x+\pi) \cos (2 x-\pi)

View answer and explanation

Correct option: B

x=\frac{n \pi}{4}+\frac{(-1)^n}{4} \sin ^{-1}\left(\frac{ \pm 2 \sqrt{2}}{3}\right)

Option B (

x=\frac{{n \pi}}{{4}}+\frac{{(-1)^n}}{{4}} \sin^{-1}(\frac{{ \pm 2 \sqrt{{2}}}}{{3}})

) is correct because simplifying the given equation leads to this general solution.

244 Easy

244 of 581

Application +4 | -1

f(\theta)=\sin ^2 \theta+\sin ^2\left(\theta+\frac{2 \pi}{3}\right)+\sin ^2\left(\theta+\frac{4 \pi}{3}\right)

, then

f\left(\frac{\pi}{15}\right)

is equal to

View answer and explanation

Correct option: B

\frac{3}{2}

Option B (

\frac{3}{2}

) is correct because simplifying the given trigonometric expression results in a constant value of

\frac{3}{2}

, independent of

\theta

245 Easy

245 of 581

Analyzing +4 | -1

The number of solutions of the equation

{2^{\cos x}} = \left| {\,\sin \,\,x\,} \right|

\left[ { - 2\;\pi ,\,\,2\;\pi } \right]

, is

View answer and explanation

Correct option: D

\mathrm{4}

Option D (4) is correct because the graphs of the two functions,

y = {2^{\cos x}}

and

y = \left| {\,\sin \,x\} \right|

, intersect at four points within the given interval

[-2\pi, 2\pi]

246 Medium

246 of 581

Analyzing +4 | -1

\cos ^2 2 x-\sin ^4 x-2 \cos ^2 x=\lambda

, has a solution

\forall \, x \in R

, then the range of

\lambda

View answer and explanation

Correct option: B

\left[-\frac{4}{3}, 0\right]

The correct answer is B (

\left[-\frac{4}{3}, 0\right]

) because it represents the range of the function after simplification and completing the square.

247 Easy

247 of 581

Understanding +4 | -1

\sin \,\,A = \sin \,\,B,\,\,\cos \,\,A = \cos \,\,B

, then the value of

A

in terms of

B

View answer and explanation

Correct option: C

2\;n\;\pi + B

Option C (

2\;n\;\pi + B

) is correct because it satisfies both the sine and cosine relations for all integers n.

248 Easy

248 of 581

Application +4 | -1

The number of solutions of

5 \cos ^2 \theta-3 \sin ^2 \theta+6 \sin \theta \cos \theta=7

in the interval

[0,2 \pi]

View answer and explanation

Correct option: C

Option C (0) is correct because by rewriting the equation using trigonometric identities, we find that the equation has no solutions in the given interval.

249 Easy

249 of 581

Application +4 | -1

The general solution of

\sin \,\,x - 3\,\,\sin \,\,2x + \sin \,\,3x = \cos \,\,x\, - 3\,\,\cos \,\,2x + \cos \,\,3x

View answer and explanation

Correct option: B

\frac{{n\pi }}{2} + \frac{\pi }{8}\;

Option B is correct because by rearranging the equation and factoring, we can find the general solution for x.

250 Easy

250 of 581

Application +4 | -1

The value of

\tan 40^{\circ}+\tan 20^{\circ}+\sqrt{3} \tan 20^{\circ} \tan 40^{\circ}

View answer and explanation

Correct option: D

\sqrt{3}

Option D (

\sqrt{3}

) is correct because the expression simplifies to

\sqrt{3}

using the tangent addition formula.

251 Easy

251 of 581

Application +4 | -1

The set of all

x

(-\pi, \pi)

satisfying

|4 \sin x-1| < \sqrt{5}

is given by

View answer and explanation

Correct option: B

\left(\frac{\pi}{10}, \frac{3 \pi}{10}\right)

Option B

\left(\frac{\pi}{10}, \frac{3 \pi}{10}\right)

is correct because solving the inequality

|4 \sin x - 1| < \sqrt{5}

for

x

in the interval

(-\pi, \pi)

yields this solution set.

252 Easy

252 of 581

Application +4 | -1

\sin \,\,\mathrm{\theta } = \sqrt 3 \,\,\cos \,\,\mathrm{\theta },\mathrm{\;\pi } < \mathrm{\theta } < 0

, then

\mathrm{\theta }

is equal to

View answer and explanation

Correct option: B

- \,\,\frac{{4\mathrm{\pi }}}{6}

Option A (

- \,\frac{{4\mathrm{\pi }}}{{6}}

) is correct because it satisfies the given equation

\sin \theta = \sqrt{3} \cos \theta

and the condition

\mathrm{\pi } < \mathrm{\theta } < 0

253 Easy

253 of 581

Analyzing +4 | -1

The solution of the equation

4{\sin ^4}x + {\cos ^4}x = 1

View answer and explanation

Correct option: D

None of the above

Option D (None of the above) is correct because the solution to the equation

4{\sin ^4}x + {\cos ^4}x = 1

x = n\pi

, which is not represented by the other options.

254 Medium

254 of 581

Understanding +4 | -1

Assertion: The inequality

a x+b y < 0

is strict inequality.

Reason: The inequality

a x+b \geq 0

is slack inequality.

View answer and explanation

Correct option: B

Assertion is correct, reason is correct; reason is not a
correct explanation for assertion

Option B (Assertion is correct, reason is correct; reason is not a correct explanation for assertion) is correct because both statements are true, but the reason doesn't explain the assertion.

255 Medium

255 of 581

Application +4 | -1

If in any

\triangle A B C

, the value of

\angle A

is obtained from the equation

3 \cos A+2=0

, then the equation whose roots are

\sin A

and

\tan A

, is

View answer and explanation

Correct option: A

6 x^2+\sqrt{5} x-5=0

Option A (

6 x^2+\sqrt{5} x-5=0

) is correct because it represents the quadratic equation whose roots are

\sin A

and

\tan A

, derived from the given trigonometric relationship.

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256 Easy

256 of 581

Application +4 | -1

The maximum and minimum values of

7 \cos \theta+24 \sin \theta

are

View answer and explanation

Correct option: A

25 and −25

Option A (25 and -25) is correct because the maximum and minimum values of

a \cos \theta + b \sin \theta

are

\sqrt{{a^2 + b^2}}

and

-\sqrt{{a^2 + b^2}}

respectively.

257 Easy

257 of 581

Analyzing +4 | -1

The number of solutions of the pair of equations

2{\sin ^2}\theta - \cos \,\,2\theta = 0

and

2{\cos ^2}\theta - 3\sin \,\,\theta = 0

in the interval

\left[ {0,\,\;2\pi } \right]

View answer and explanation

Correct option: C

Two

Option C (Two) is correct because solving the given equations yields

\sin \theta = \frac{1}{2}

, which has two solutions in the interval

[0, 2\pi]

258 Medium

258 of 581

Analyzing +4 | -1

The general solution of

\sin 3 x+\sin x-3 \sin 2 x

=\cos 3 x+\cos x-3 \cos 2 x

View answer and explanation

Correct option: A

\frac{n \pi}{2}+\frac{\pi}{8}

for

n

integer

Option A is correct because the general solution is found by simplifying the trigonometric equation and solving for x.

259 Medium

259 of 581

Analyzing +4 | -1

The number of

x \in[0,2 \pi]

for which

\left|\sqrt{2 \sin ^{4} x+18 \cos ^{2} x}-\sqrt{2 \cos ^{4} x+18 \sin ^{2} x}\right|=1 \text { is }

View answer and explanation

Correct option: D

Option D (8) is correct because squaring both sides and solving the equation gives 8 solutions in the interval [0, 2

\pi

260 Easy

260 of 581

Application +4 | -1

5\,\,\cos \,\,2\;\theta + 2\,\,{\cos ^2}\,\,\frac{\theta }{2} + 1 = 0,\, - \pi < \theta < \pi

, then

\theta =

View answer and explanation

Correct option: D

\frac{\pi }{3},\,\,\pi - {\cos ^{ - 1}}\,\,\left( {3/5} \right)

Option D (

\frac{\pi }{3},\,\pi - {\cos ^{ - 1}}\,\,\left( {3/5} \right)

) is correct because these are the solutions to the given trigonometric equation within the specified interval.

261 Medium

261 of 581

Analyzing +4 | -1

2 \tan ^2 x-5 \sec x

is equal to 1 for exactly 7 distinct values of

x \in\left[0, \frac{n \pi}{2}\right] n \in N

, then the greatest value of

n

View answer and explanation

Correct option: D

Option D (15) is correct because the equation simplifies to sec(x) = 3, which yields 7 distinct solutions within the given interval, resulting in n=15.

262 Medium

262 of 581

Application +4 | -1

Find the value of

\cos (29 \pi / 3)

View answer and explanation

Correct option: D

1 / 2

Option D (1/2) is correct because by simplifying the angle and using trigonometric identities,

\cos (29 \pi / 3)

can be reduced to

\cos (\pi/3)

which equals 1/2.

263 Medium

263 of 581

Application +4 | -1

If the solutions for

\mathrm{\theta },\,\cos p\mathrm{\theta } + \cos q\mathrm{\theta } = 0,\;\,p > 0,\;\,q > 0

are in AP, then the numerically smallest common difference of AP is

View answer and explanation

Correct option: B

\frac{{2\pi }}{{p + q}}

Option B (

\frac{{2\pi }}{{p + q}}

) is correct because it represents the smallest common difference of the arithmetic progression formed by the solutions of the given trigonometric equation.

264 Easy

264 of 581

Application +4 | -1

2\,\,{\sin ^2}\theta + \sqrt 3 \,\,\cos \,\,\theta + 1 = 0

, then the value of

\theta

View answer and explanation

Correct option: C

\frac{{5\,\pi }}{6}

Option C is correct because it is the solution to the given trigonometric equation within the possible range of values.

265 Easy

265 of 581

Analyzing +4 | -1

\left( {\sec \,\,\theta - 1} \right) = \left( {\sqrt 2 - 1} \right)\tan \,\,\theta

, then

\theta =

View answer and explanation

Correct option: B

2\;n\;\pi ,\,\,2\;n\;\pi + \frac{\pi }{4},\,\,n \in Z

Option B (

2\;n\;\pi ,\,2\;n\;\pi + \frac{\pi }{4},\,n \in Z

) is correct because it represents the general solutions obtained by solving the given trigonometric equation.

266 Easy

266 of 581

Analyzing +4 | -1

The minimum value of the expression

\sin \alpha+\sin \beta+\sin \gamma

, where

\alpha, \beta

and

\gamma

are real numbers satisfying

\alpha+\beta+\gamma=\pi

, is

View answer and explanation

Correct option: C

Negative

Option C (Negative) is correct because the minimum value of the expression can be negative when α, β, and γ are real numbers satisfying α+β+γ=π.

267 Medium

267 of 581

Application +4 | -1

The general value of

x

for which

\cos \,\,2x,\,\,\frac{1}{2}\;\,\,\mathrm{and}\,\,\sin \,\,2x

are in AP, are given by

View answer and explanation

Correct option: B

n\pi ,\;\,n\pi + \frac{\pi }{4}

Option B (-1) is correct because the sum telescopes to

\cos \pi

which equals -1.

268 Easy

268 of 581

Analyzing +4 | -1

The equation

\sin \,\,x\,\,\cos \,\,x = 2

has

View answer and explanation

Correct option: D

No solution

Option D (No solution) is correct because the equation simplifies to

\sin 2x = 4

, which is impossible since the range of the sine function is [-1, 1].

269 Medium

269 of 581

Application +4 | -1

0 \leq x \leq \frac{\pi}{2}

and

81^{\sin ^2 x}+81^{\cos ^2 x}=30

, then

x

is equal to

View answer and explanation

Correct option: A

\frac{\pi}{6}, \frac{\pi}{3}

Option A (

\frac{\pi}{6}, \frac{\pi}{3}

) is correct because solving the given equation by substituting

y = 81^{\sin^2 x}

leads to finding the values of

\sin x

and hence

x

270 Medium

270 of 581

Application +4 | -1

The solution of

\tan x+\tan 2 x+\tan 3 x=\tan x \tan 2 x \tan 3 x

View answer and explanation

Correct option: C

x=\frac{n \pi}{3}, n \in I

Option C (

x=\frac{{n \pi}}{{3}}, n \in I

) is correct because it is derived by using the tangent addition formula and simplifying the given equation.

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