The general solution of ${\sin ^2}\,\,{\rm{\theta }}\,\,\sec \,\,{\rm{\theta }} + \sqrt 3 \,\,\tan \,\,{\rm{\theta }} = 0$ is

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

${\rm{\theta }} = n\pi + {\left( { - 1} \right)^{n + 1}}\,\,\frac{\pi }{3},\,\,n \in I$

${\rm{\theta }} = \frac{{n\pi }}{2},\,\;n \in I$

If $\sin ^2 x+\cos ^2 y=2 \sec ^2 z$, then

$x=(m+1) \frac{\pi}{2}, y=2 n \pi$ and $z=t \pi$

$x=(2 m-1) \frac{\pi}{2}, y=n \pi$ and $z=2 t \pi$

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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 391-420 of 581 questions

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391 Easy

391 of 581

Application +4 | -1

\tan \left( {k + 1} \right)\theta = \tan \,\,\theta

, then

\theta

belongs to the set

View answer and explanation

Correct option: C

\left\{ {n\pi \;/\;k\;:n \in I} \right\}

Option C (

\{n\pi \;/\;k\;:\;n \in I\}

) is correct because it represents the general solution for

\theta

derived from the given trigonometric equation.

392 Medium

392 of 581

Analyzing +4 | -1

\cot \theta \,\,\cot 7\theta + \cot \theta \,\,\cot 4\theta + \cot 4\theta \,\,\cot 7\theta = 1

then

\theta =

View answer and explanation

Correct option: D

\frac{{n\;\pi }}{{12}},\,\,n \in Z

Option D (

\frac{{n\\;\pi }}{{12}},\\,n \in Z

) is correct because it represents the general solution to the given trigonometric equation.

393 Medium

393 of 581

Application +4 | -1

The most general value of

\theta

for which

\sin \,\,\theta - \cos \,\,\theta = \mathop {\min }\limits_{x\; \in \;R} \,\,\left\{ {1,\;\,{x^2} - 4x + 6} \right\}

are given by

View answer and explanation

Correct option: B

\theta = n\pi \,\, + \,\,{\left( { - 1} \right)^n}\frac{\pi }{4}\,\, + \,\,\frac{\pi }{4},\,\,n \in Z

Option B (

\theta = n\pi \\ + \\ {\left( { - 1} \right)^n}\frac{\pi }{4}\\ + \\ \frac{\pi }{4},\n \in Z

) is correct because it is the general solution of the trigonometric equation after finding the minimum value of the quadratic expression.

394 Hard

394 of 581

Analyzing +4 | -1

The number of ordered pairs

\left( {x,y} \right)

satisfying

y = 2\sin \,\,x\;\,\mathrm{and}\,\;y = 5{x^2} + 2x + 3

View answer and explanation

Correct option: A

Option A (0) is correct because the two equations

y = 2\sin x

and

y = 5x^2 + 2x + 3

have no intersection points, hence no solutions.

395 Medium

395 of 581

Application +4 | -1

\sin x \cdot \sin \left(60^{\circ}-x\right) \cdot \sin \left(60^{\circ}+x\right)=1 / 8

and

n \in \mathbb{Z}

, then:

View answer and explanation

Correct option: B

x=\frac{n \pi}{3}+(-1)^n \frac{\pi}{18}

Option B (

x=\frac{{n \pi}}{{3}}+(-1)^n \frac{{\pi}}{{18}}

) is correct because it is derived by using the trigonometric identity and solving the resulting sine equation.

396 Medium

396 of 581

Understanding +4 | -1

The equation

p \cos x - q \sin x = r

admits of a solution for

x

only, if

View answer and explanation

Correct option: B

-\sqrt{p^2+q^2} < r < \sqrt{p^2+q^2}

The correct answer is B because the equation has a solution only if

r

lies within the range of the expression, which is

-\sqrt{p^2+q^2} \leq r \leq \sqrt{p^2+q^2}

397 Medium

397 of 581

Analyzing +4 | -1

The smallest positive number

p

for which the equation

\cos (p \sin x)=\sin (p \cos x)

has a solution

x \in[0,2 \pi]

, is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the smallest positive value of

p

that satisfies the given equation.

398 Easy

398 of 581

Application +4 | -1

\frac{3 \cos \theta+\cos 3 \theta}{3 \sin \theta-\sin 3 \theta}

is equal to

View answer and explanation

Correct option: C

\cot ^3 \theta

Option C (

\cot^3 \theta

) is correct because by applying the triple angle formulas for cosine and sine, the expression simplifies to

\cot^3 \theta

399 Easy

399 of 581

Analyzing +4 | -1

The general solution of

{\sin ^2}\,\,\mathrm{\theta }\,\,\sec \,\,\mathrm{\theta } + \sqrt 3 \,\,\tan \,\,\mathrm{\theta } = 0

View answer and explanation

Correct option: B

\mathrm{\theta } = n\pi ,\;\,n \in I

Option B (θ = nπ, n ∈ I) is correct because the equation simplifies to tan θ (sin θ + √3) = 0, and since sin θ + √3 ≠ 0, we have tan θ = 0, which gives θ = nπ.

400 Medium

400 of 581

Application +4 | -1

\alpha+\beta=\frac{\pi}{2}

and

\beta+\gamma=\alpha

, then

\tan \alpha

is equal to

View answer and explanation

Correct option: C

\tan \beta+2 \tan \gamma

Option C (

\tan \beta+2 \tan \gamma

) is correct because it expresses

\tan \alpha

in terms of

\tan \beta

and

\tan \gamma

based on the given relationships between the angles.

401 Medium

401 of 581

Analyzing +4 | -1

The set of values of

x

for which

\frac{{\tan \,\,3\;x - \tan \,\,2\;x}}{{1 + \tan \,\,3\;x\,\,\tan \,\,2\;x}} = 1

View answer and explanation

Correct option: A

\mathrm{\Phi }

Option A (

\Phi

) is correct because the solution obtained from simplifying the equation makes the original equation undefined, thus there is no solution.

402 Medium

402 of 581

Application +4 | -1

2 \cos x < \sqrt{3}

and

x \in[-\pi, \pi]

, then the solution set for

x

View answer and explanation

Correct option: A

\left[-\pi,-\frac{\pi}{6}\right) \cup\left(\frac{\pi}{6}, \pi\right]

Option A ([-π,-

\frac{π}{6}

) ∪ (

\frac{π}{6}

, π]) is correct because it represents the interval where

\cos x < \frac{{\sqrt{{3}}}}{{2}}

within the given range of

x \in [-\pi, \pi]

403 Easy

403 of 581

Analyzing +4 | -1

The possible values of

\theta \in(0, \pi)

such that

\sin (\theta)+\sin (4 \theta)+\sin (7 \theta)=0

are

View answer and explanation

Correct option: D

\frac{2 \pi}{9}, \frac{\pi}{4}, \frac{4 \pi}{9}, \frac{\pi}{2}, \frac{3 \pi}{4}, \frac{8 \pi}{9}

The correct answer is D because it lists all the values of

\theta \in (0, \pi)

that satisfy the given trigonometric equation.

404 Medium

404 of 581

Analyzing +4 | -1

Solution of the equation

{\cos ^2}\left( {\frac{1}{2}px} \right) + {\cos ^2}\left( {\frac{1}{2}qx} \right) = 1

form an arithmetic progression with common difference

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the solution forms two arithmetic progressions with different common differences, neither of which matches the provided options.

405 Medium

405 of 581

Analyzing +4 | -1

Statement - 1:

\frac{\tan 3 x-\tan 2 x}{1+\tan 3 x \tan 2 x}=1, x=n \pi+\frac{\pi}{4}, n \in I

Statement - 2:

\tan {x}

is not defined at

x=n \pi+\frac{\pi}{2}, n \in I

View answer and explanation

Correct option: D

Statement-1 is False, Statement-2 is True

Statement-1 is False, Statement-2 is True because the given equation simplifies to tan(x) = 1, but tan(2x) is undefined at x = nπ + π/4.

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406 Medium

406 of 581

Application +4 | -1

The general solution for

\alpha

is, if

1 / 6 \sin \alpha, \cos \alpha, \tan \alpha

in G.P.

View answer and explanation

Correct option: C

2 n \pi \pm \frac{\pi}{3}

Option C (

2 n \pi \pm \frac{{\pi}}{{3}}

) is correct because it represents the general solution derived from the given geometric progression condition and solving for

\alpha

407 Medium

407 of 581

Analyzing +4 | -1

The number of elements in the set

S=\left\{\theta \in[0,2 \pi]: 3 \cos ^4 \theta-5 \cos ^2 \theta-2 \sin ^6\theta+2=0\right\}

View answer and explanation

Correct option: D

Option C (Both the statements (A) and (B) are true) is correct because the given inequality implies that f(x) is bounded between 1/8 and 1 for all x in the interval [2,4].

408 Medium

408 of 581

Application +4 | -1

a=\cos \phi \cos \psi+\sin \phi \sin \psi \cos \delta

b=\cos \phi \sin \psi-\sin \phi \cos \psi \cos \delta

and

c=\sin \phi \sin \delta

. Then

a^2+b^2+c^2=

View answer and explanation

Correct option: C

Option C (1) is correct because by expanding the expression a² + b² + c² and using trigonometric identities, it simplifies to 1.

409 Easy

409 of 581

Understanding +4 | -1

|x| < 3

and

x

is a real number, then

-m < x < m

. The value of

m

View answer and explanation

Correct option: A

Option A (3) is correct because the definition of absolute value implies that if

|x| < 3

, then

-3 < x < 3

, thus

m = 3

410 Easy

410 of 581

Application +4 | -1

Let

-\frac{\pi}{6} < \theta < -\frac{\pi}{12}

. suppose

\alpha_1

and

\beta_1

are the roots of the equation

x^2-2 x \sec \theta+1=0

and

\alpha_2, \beta_2

are the roots of the equation

x^2+2 x \tan \theta-1=0

. If

\alpha_1 > \beta_1

and

\alpha_2 > \beta_2

, then

\alpha_1+\beta_2

equals

View answer and explanation

Correct option: C

-2 \tan \theta

Option C (-2 tan θ) is correct because it is derived by finding the roots of the given equations and summing α_1 and β_2.

411 Easy

411 of 581

Application +4 | -1

The solution of the equation

\tan 2 \theta \tan \theta=1

View answer and explanation

Correct option: A

\theta=n \pi \pm \frac{\pi}{6}

Option A (

\theta=n \pi \pm \frac{{\pi}}{{6}}

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

412 Medium

412 of 581

Analyzing +4 | -1

The number of solutions of the equation

16\left(\sin ^5 x+\cos ^5 x\right)=11(\sin x+\cos x)

in the interval

[0,2 \pi]

View answer and explanation

Correct option: A

Option A (6) is correct because the equation simplifies to a trigonometric equation with 6 solutions in the given interval.

413 Easy

413 of 581

Application +4 | -1

\alpha ,\;\,\beta \in \left( {0,\;\,\frac{\pi }{2}} \right),\,\,\sin \,\,\alpha = \frac{4}{5}

and

\cos \,\,\left( {\alpha + \beta } \right) = - \frac{{12}}{{13}}

, then

\sin \,\,\beta

is equal to

View answer and explanation

Correct option: A

\frac{{63}}{{65}}

Option A,

\frac{{63}}{{65}}

, is correct because it is the value of

\sin \,\beta

calculated using the given values of

\sin \,\alpha

and

\cos \,\left( {\alpha + \beta } \right)

and the trigonometric identity for

\sin(\alpha + \beta - \alpha)

414 Medium

414 of 581

Application +4 | -1

\frac{\pi}{2} < \alpha < \pi, \pi < \beta < \frac{3 \pi}{2} ; \sin \alpha=\frac{15}{17}

and

\tan \beta=\frac{12}{5}

, then the value of

\sin (\beta-\alpha)

View answer and explanation

Correct option: D

\frac{171}{221}

The correct answer is

\frac{171}{221}

because it is the result of applying the sine subtraction formula and using the given information to find the values of sine and cosine.

415 Medium

415 of 581

Analyzing +4 | -1

\sin ^2 x+\cos ^2 y=2 \sec ^2 z

, then

View answer and explanation

Correct option: A

x=(2 m+1) \frac{\pi}{2}, y=n \pi

and

z=t \pi

Option A (

x=(2 m+1) \frac{\pi}{2}, y=n \pi

and

z=t \pi

) is correct because it satisfies the given equation

\sin ^2 x+\cos ^2 y=2 \sec ^2 z

when each term reaches its extreme value.

416 Easy

416 of 581

Application +4 | -1

Let

A=\sin ^4 x+\cos ^4 x, x \in \mathbf{R}

, then

View answer and explanation

Correct option: A

\frac{1}{2} \leq A \leq 1

The correct answer is A (

\frac{1}{2} \leq A \leq 1

) because by simplifying the expression and using the range of sin²(2x), we can find the range of A.

417 Medium

417 of 581

Application +4 | -1

Suppose

\theta \in\left[0, \frac{\pi}{4}\right]

is a solution of

4 \cos \theta-3 \sin \theta=1.

Then

\cos \theta

is equal to:

View answer and explanation

Correct option: A

\frac{4}{(3 \sqrt{6}-2)}

The correct answer is D (

\frac{{4}}{{(3 \sqrt{{6}}-2)}}

) because it is the value of cos(theta) obtained by solving the trigonometric equation and using the half-angle substitution.

418 Easy

418 of 581

Analyzing +4 | -1

f(\theta)=\sin ^4 \theta+\cos ^4 \theta+1

, then range of

f(\theta)

View answer and explanation

Correct option: A

\left[\frac{3}{2}, 2\right]

Option A (

\left[\frac{3}{2}, 2\right]

) is correct because it represents the range of the function after simplification using trigonometric identities.

419 Medium

419 of 581

Application +4 | -1

The value of

\sum\limits_{{r=0}}^{10} \cos ^3 \frac{\pi r}{3}

View answer and explanation

Correct option: D

-\frac{1}{8}

Option D (-

\frac{1}{8}

) is correct because by using the trigonometric identity

\cos^3 x = \frac{1}{4}(\cos 3x + 3 \cos x)

and simplifying the summation, the value is found to be -

\frac{1}{8}

420 Medium

420 of 581

Analyzing +4 | -1

3 \sin ^2 \theta+2 \sin ^2 \phi=1

and

3 \sin 2 \theta=2 \sin 2 \phi, 0 < \theta < \frac{\pi}{2}

and

0 < \phi < \frac{\pi}{2}

, then the value of

\theta+2 \phi

View answer and explanation

Correct option: A

\frac{\pi}{2}

Option A (

\frac{\pi}{2}

) is correct because after simplification and using trigonometric identities,

\theta + 2\phi

equals

\frac{\pi}{2}

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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