If $0 \le x \le \pi $ and ${81^{{{\sin }^2}x}} + \,\,{81^{{{\cos }^2}x}} = 30$, then $x$ is equal to

$\frac{\pi }{3},\,\,\frac{\pi }{2}$

$\frac{{5\,\pi }}{6},\,\,\frac{\pi }{3}$

$\frac{{2\,\pi }}{3},\,\,\frac{\pi }{3}$

The most general values of ${\rm{\theta }}$ satisfying $\tan \,\,{\rm{\theta }} + \tan \,\,\left( {\frac{{3\,\pi }}{4} + {\rm{\theta }}} \right) = 2$ ${\rm{is}}\,{\rm{/}}\,{\rm{are}}$

$2n\pi + \frac{\pi }{3},\;\,n \in I$

$2n\pi \,\, \pm \,\,\frac{\pi }{3},\;\,n \in I$

$2n\pi \,\, + \,\,{\left( { - 1} \right)^n}\,\,\frac{\pi }{3},\,\;n \in I$

The solution of $\sin \,\,x + \sin \,\,5x = \sin \,\,3x$ in $\left( {0,\;\,\frac{\pi }{2}} \right)$

$\frac{\pi }{4},\,\;\frac{\pi }{{10}}$

$\frac{\pi }{4},\;\,\frac{\pi }{2}$

$\frac{\pi }{8},\;\,\frac{\pi }{{16}}$

If $f(\theta)=\frac{1}{2-3 \cos \theta}$, then range of $f(\theta)$ is

$\left(-\infty, \frac{1}{5}\right] \cup[1, \infty)$

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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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451 Easy

451 of 581

Application +4 | -1

0 \le x \le \pi

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 these values satisfy the given trigonometric equation within the specified interval.

452 Easy

452 of 581

Application +4 | -1

The most general values of

\mathrm{\theta }

satisfying

\tan \,\,\mathrm{\theta } + \tan \,\,\left( {\frac{{3\,\pi }}{4} + \mathrm{\theta }} \right) = 2

\mathrm{is}\,\mathrm{/}\,\mathrm{are}

View answer and explanation

Correct option: A

n\pi \,\, \pm \,\,\frac{\pi }{3},\,\,n \in I

Option A (

n\pi \\; \pm \\; \frac{\pi }{{3}},\\;n \in I

) is correct because it represents the general solution of the trigonometric equation after simplification.

453 Medium

453 of 581

Analyzing +4 | -1

Let

f: R \rightarrow R

be a function defined by

f(x)=\log _{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}

, for some

m

, such that the range of

f

[0,2]

. Then the value of

m

View answer and explanation

Correct option: D

Option D (5) is correct because the range of the function is [0,2], which leads to the equation

\log_{\sqrt{m}}(m-4) = 0

, and solving for m gives m = 5.

454 Hard

454 of 581

Application +4 | -1

\theta

lies in the first quadrant and

\cos \theta

=8 / 17

, then the value of

\cos \left(30^{\circ}+\theta\right)+\cos \left(45^{\circ}-\theta\right)+

\cos \left(120^{\circ}-\theta\right)

View answer and explanation

Correct option: A

\left(\frac{\sqrt{3}-1}{2}+\frac{1}{\sqrt{2}}\right) \frac{23}{17}

Option A is correct. By using trigonometric identities and the given value of

\cos \theta

, the expression simplifies to

\left(\frac{{\sqrt{{3}}-1}}{{2}}+\frac{{1}}{{\sqrt{{2}}}}\right) \frac{{23}}{{17}}

455 Easy

455 of 581

Application +4 | -1

The most general value of

\theta

satisfying

\tan \,\,\theta + \tan \left( {\frac{{3\;\pi }}{4} + \theta } \right) = 2\;\,\,are

View answer and explanation

Correct option: A

n\;\pi \,\, \pm \,\,\frac{\pi }{3},\,\,n \in Z

Option A (

n\\;\pi \\, \pm \\, \frac{\pi }{{3}},\\,n \in Z

) is correct because solving the trigonometric equation leads to

\tan^2 \theta = 3

, which gives the general solution

\theta = n\pi \pm \frac{\pi}{3}

456 Medium

456 of 581

Analyzing +4 | -1

\tan 81^{\circ}-\tan 63^{\circ}-\tan 27^{\circ}+\tan 9^{\circ}

equals

View answer and explanation

Correct option: D

Option D (4) is correct because simplifying the expression using trigonometric identities leads to the value 4.

457 Easy

457 of 581

Analyzing +4 | -1

If the values of

x

and

y

satisfy the equation

\tan ^2(x+y)+\cot ^2(x+y)=1-2 x-x^2

, then the value of

x^2-3 x+2

is _____.

View answer and explanation

Correct option: A

Option C (3) is incorrect. The correct answer is 6 because the given equation is valid only when both sides are equal to 2, leading to x = -1, and thus

x^2 - 3x + 2 = 6

458 Easy

458 of 581

Application +4 | -1

The number of solutions of

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

between

\frac{{ - \pi }}{2}

and

\frac{\pi }{2}

View answer and explanation

Correct option: A

The correct answer is A (3) because solving the trigonometric equation

\sin x = \sin 2x

within the given interval yields three solutions: 0,

\frac{\pi}{3}

, and

-\frac{\pi}{3}

459 Easy

459 of 581

Application +4 | -1

The solution of the equation

\cos ^2 \theta+\sin \theta+1=0

lies in the interval

View answer and explanation

Correct option: D

None of these

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

\cos^2 \theta + \sin \theta + 1 = 0

\theta = 2n\pi - \frac{\pi}{2}

, and none of the given intervals contain this solution.

460 Easy

460 of 581

Understanding +4 | -1

The equation

\sin x+\sin y+\sin z=-3

for

0 \leq x \leq 2 \pi

0 \leq y \leq 2 \pi, 0 \leq z \leq 2 \pi

, has

View answer and explanation

Correct option: A

One solution

Option A (One solution) is correct because the maximum value of

\sin x

is 1, so for the sum to be -3, each term must be -1, leading to a unique solution.

461 Easy

461 of 581

Application +4 | -1

The solution of

\sin \,\,x + \sin \,\,5x = \sin \,\,3x

\left( {0,\;\,\frac{\pi }{2}} \right)

View answer and explanation

Correct option: B

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

Option B (

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

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

462 Medium

462 of 581

Analyzing +4 | -1

\cos x=\tan y, \cot y=\tan z

and

\cot z=\tan x

, then

\sin x

is equal to

View answer and explanation

Correct option: D

\frac{\sqrt{5}-1}{2}

Option D (

\frac{\sqrt{5}-1}{2}

) is correct because substituting the given relations leads to a quadratic equation in

\sin x

, and solving it yields this value as the valid solution.

463 Easy

463 of 581

Application +4 | -1

The smallest positive angle which satisfies the equation

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

View answer and explanation

Correct option: A

\frac{5 \pi}{6}

Option A (

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

) is correct because it is the smallest positive angle that satisfies the given trigonometric equation after simplification and solving for

\theta

464 Medium

464 of 581

Application +4 | -1

The number of solutions of

16^{\sin ^2 x}+16^{\cos ^2 x}=10,0 \leq x \leq 2 \pi

View answer and explanation

Correct option: A

Option A (8) is correct because solving the equation

16^{\sin^2 x} + 16^{\cos^2 x} = 10

leads to

\sin x = \pm \frac{1}{2}

and

\sin x = \pm \frac{\sqrt{3}}{2}

, which have a total of 8 solutions in the interval

[0, 2\pi]

465 Medium

465 of 581

Application +4 | -1

\tan ({A}+{B})=3 \tan {A}

, then

\frac{\sin 2({A}+{B})+\sin 2 {A}}{\sin 2 {B}}=

View answer and explanation

Correct option: A

Option A (2) is correct because it represents the value of the expression

\frac{{\sin 2(A+B)+\sin 2 A}}{{\sin 2 B}}

given the condition

\tan (A+B)=3 \tan A

, which simplifies to 2 using trigonometric identities.

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466 Easy

466 of 581

Analyzing +4 | -1

\tan (\pi \cos \theta)=\cot (\pi \sin \theta)

, then the value(s) of

\cos \left(\theta-\frac{\pi}{4}\right)

is/are

View answer and explanation

Correct option: C

\pm \frac{1}{2 \sqrt{2}}

Option C,

\pm \frac{{1}}{{2 \sqrt{{2}}}}

is correct because it represents the possible values of

\cos \left(\theta-\frac{{\pi}}{{4}}\right)

obtained by solving the given trigonometric equation.

467 Medium

467 of 581

Application +4 | -1

-\pi \leq x \leq \pi,-\pi \leq y \leq \pi

and

\cos x+\cos y=2

, then the value of

\cos (x-y)

View answer and explanation

Correct option: C

Option C (1) is correct because the maximum value of

\cos x

is 1, so both

\cos x

and

\cos y

must be 1, leading to

x=0

and

y=0

, and thus

\cos(x-y) = 1

468 Easy

468 of 581

Application +4 | -1

1+\cot \theta \leq \frac{\pi}{2}

, for

0 < \theta < \pi

, then

\theta

is equal to

View answer and explanation

Correct option: C

\frac{\pi}{2}

Option C (

\frac{{\pi}}{{2}}

) is correct because it satisfies the inequality

1 + \cot \theta \le \cot \frac{{\theta}}{{2}}

for

0 < \theta < \pi

469 Easy

469 of 581

Application +4 | -1

\sin \alpha \sin \beta-\cos \alpha \cos \beta+1=0

, then the value of

\cot \alpha \tan \beta

View answer and explanation

Correct option: A

-1

Option A (-1) is correct because the given condition implies that

\alpha + \beta = 2n\pi

, and thus

\cot \alpha \tan \beta = -1

470 Easy

470 of 581

Application +4 | -1

x=\tan ^{-1} 3+\tan ^{-1} 2

\,\,y=\tan ^{-1} 3-\tan ^{-1} 2

a \tan x+b \tan y=0

, then

a+b

is equal to

k a

where

k

is equal to

View answer and explanation

Correct option: D

The correct answer is D (8) because solving for

\tan x

and

\tan y

and substituting into the given equation allows us to find the value of k.

471 Medium

471 of 581

Application +4 | -1

m=\operatorname{cosec} \theta-\sin \theta

and

n=\sec \theta-\cos \theta

, then

\mathrm{m}^{2 / 3}+\mathrm{n}^{2 / 3}=

View answer and explanation

Correct option: A

(\mathrm{mn})^{-2 / 3}

Option C (

(mn)^{-2 / 3}

) is correct because simplifying the expressions for m and n and then substituting them into the target expression yields the result.

472 Medium

472 of 581

Analyzing +4 | -1

The sum of the series

\sin \theta \sec 3 \theta+\sin 3 \theta \sec 3^2 \theta+\sin 3^2 \theta \sec 3^3 \theta

+\ldots n

terms, is

View answer and explanation

Correct option: C

\frac{1}{2}\left[\tan 3^n \theta-\tan \theta\right]

Option C,

\frac{1}{2}\left[\tan 3^n \theta-\tan \theta\right]

, is correct because the series telescopes, leaving only the first and last terms after simplification.

473 Medium

473 of 581

Application +4 | -1

The sum of all the solutions of the equation

\cos x \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \cos \left(\frac{\pi}{3}-x\right)=\frac{1}{4}, x \in[0,6 \pi]

View answer and explanation

Correct option: C

\frac{110 \pi}{3}

None of the options are correct. The sum of all solutions is

30\pi

, so Option D (None of these) is correct.

474 Medium

474 of 581

Application +4 | -1

2 \tan ^2 \theta=\sec ^2 \theta

, then the general value of

\theta

View answer and explanation

Correct option: C

n \pi \pm \frac{\pi}{4}

Option C (

n \pi \pm \frac{{\pi}}{{4}}

) is correct because it represents the general solution derived from solving the given trigonometric equation.

475 Medium

475 of 581

Application +4 | -1

Let

n

be a positive integer such that

\sin \frac{\pi}{2 n}+\cos \frac{\pi}{2 n}=\frac{\sqrt{n}}{2}

, then

View answer and explanation

Correct option: D

4 < n < 8

Option D (

4 < n < 8

) is correct because by squaring the given equation and substituting n=6, the equation is satisfied.

476 Easy

476 of 581

Application +4 | -1

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

, then the value of

\cos \,\,\left( {\theta + \frac{\pi }{4}} \right)

equals

View answer and explanation

Correct option: B

\frac{1}{{2\,\sqrt 2 }}

Option B (

\frac{1}{2\sqrt 2 }

) is correct because by using trigonometric identities, the equation simplifies to

\cos(\theta + \frac{\pi}{4}) = \frac{1}{2\sqrt 2 }

477 Easy

477 of 581

Application +4 | -1

\cot \,\,\theta = \sin \,\,2\;\theta ,\,\,\theta \ne n\pi ,\,\,n \in Z

, if

\theta

equals

View answer and explanation

Correct option: A

45^\circ

90^\circ

Option A (

45^\circ

90^\circ

) is correct because these are the solutions to the equation

\cot \theta = \sin 2\theta

within the given constraints.

478 Medium

478 of 581

Analyzing +4 | -1

The equation

e^{|\sin x|}+e^{-|\sin x|}+4 k=0

will have exactly four distinct solutions in

[0,2 \pi]

View answer and explanation

Correct option: D

none of these

Option D (none of these) is correct because the given equation has no solution for k in the specified ranges.

479 Medium

479 of 581

Application +4 | -1

f(\theta)=\frac{1}{2-3 \cos \theta}

, then range of

f(\theta)

View answer and explanation

Correct option: C

(-\infty,-1] \cup\left(\frac{1}{5}, \infty\right)

Option D (

( -\infty, -1] \cup (\frac{1}{5}, \infty)

) is correct because by analyzing the range of

2-3\cos \theta

, we find the range of

f(\theta)

to be

(-\infty, -1] \cup [\frac{1}{5}, \infty)

excluding 0.

480 Medium

480 of 581

Analyzing +4 | -1

The number of values of

\theta

in the interval

\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

satisfying the equation

(1-\tan \theta)(1+\tan \theta) \sec ^2 \theta+2^{\tan ^2 \theta}=0

View answer and explanation

Correct option: C

Option D (None of these) is correct because the equation simplifies to a form where there are no solutions for θ in the given interval.

Revision notes and key takeaways

Review TRIGONOMETRIC EQUATIONS in your textbook or class notes, then use this set to check your understanding.

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