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

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

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

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

Solution to the equation $\tan p \theta-\cot q \theta$ is

$\frac{(n+2)}{(p+q)} \pi \cdot n \in I$

$(2 n+1) \cdot \frac{\pi}{2}, n \in I$

If $\cos \,\,{\rm{\theta }} = - \frac{1}{{\sqrt {2\;} }}$ and $\tan \,\,{\rm{\theta }} = 1$, then the general value of ${\rm{\theta }}$ is

$2\left( {n + 1} \right){\rm{\pi }} + \frac{{\rm{\pi }}}{4}$

$2n{\rm{\pi }} + \frac{{\rm{\pi }}}{4}$

$n{\rm{\pi }} + \frac{{\rm{\pi }}}{4}$

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

If $\cos \,\,p\;\theta = \cos \,\,q\;\theta ,\,\,p \ne q$ then

$\theta = 2\;n\;\pi ,\,\,n \in Z$

$\theta = \frac{{n\;\pi }}{{p + q}},\,\,n \in Z$

If the distance between the points $\left( {a\cos {\rm{\theta }},a\sin {\rm{\theta }}} \right)$ and $\left( {a\cos \phi ,a\sin \phi } \right)$ is $2a$, then ${\rm{\theta }}$ is equal to

$n\pi + \frac{\pi }{2} + \phi ,n \in Z$

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 $\tan \,\,{\rm{\theta }} + \tan \,\,\left( {\frac{{3\,\pi }}{4} + {\rm{\theta }}} \right) = 2$ ${\rm{is}}\,{\rm{/}}\,{\rm{are}}$

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

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

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

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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Easy

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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 181-210 of 261 questions

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181 Easy

181 of 261

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

182 Easy

182 of 261

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.

183 Easy

183 of 261

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.

184 Easy

184 of 261

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

185 Easy

185 of 261

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.

186 Easy

186 of 261

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.

187 Easy

187 of 261

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)

188 Easy

188 of 261

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.

189 Easy

189 of 261

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.

190 Easy

190 of 261

Analyzing +4 | -1

In a

\triangle P Q R

, if

3 \sin P+4 \cos Q=6

and

4 \sin Q+3 \cos P=1

, then

\angle R

is equal to

View answer and explanation

Correct option: B

\frac{\pi}{6}

Option B (

\frac{\pi}{6}

) is correct because solving the system of trigonometric equations and considering the possible values for angles in a triangle leads to this result.

191 Easy

191 of 261

Application +4 | -1

Solution to the equation

\tan p \theta-\cot q \theta

View answer and explanation

Correct option: A

\frac{(2 n+1)}{(p+q)} \frac{\pi}{2}, n \in I

Option A is correct because the solution is derived by simplifying the given trigonometric equation and solving for

\theta

192 Easy

192 of 261

Analyzing +4 | -1

The number of solutions of the equation

\tan \,\,x + \sec \,\,x = 2\,\,\cos \,\,x

and

\cos \,\,x \ne 0

lying in the interval

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

View answer and explanation

Correct option: A

\mathrm{2}

Option A (2) is correct because the equation has two solutions in the given interval after considering the domain restrictions.

193 Easy

193 of 261

Application +4 | -1

\frac{\cos A}{3}=\frac{\cos B}{4}=\frac{1}{5}-\frac{\pi}{2} < A < 0

and

-\frac{\pi}{2} < B < 0

, then the value of

2 \sin A+4 \sin B

View answer and explanation

Correct option: C

-4

Option C (-4) is correct because it is the result of calculating

2 \sin A + 4 \sin B

using the given conditions and quadrant information.

194 Easy

194 of 261

Application +4 | -1

A root of the equation

\sin x+x-1=0

lies in the interval

View answer and explanation

Correct option: A

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

Option A

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

is correct because the equation

\sin x + x - 1 = 0

has a root in this interval, which can be seen by observing the behavior of the function at the interval's endpoints.

195 Easy

195 of 261

Application +4 | -1

\cot \,\,x + \mathrm{cose}c\;\,\,x = \sqrt 3

, then the principle value of

\left( {x - \frac{\pi }{6}} \right)

View answer and explanation

Correct option: D

\frac{\pi }{6}

Option D (

\frac{\pi }{6}

) is correct because solving the given trigonometric equation

\cot x + \csc x = \sqrt{3}

leads to

x = 2n\pi + \frac{\pi}{3}

, and substituting into

x - \frac{\pi}{6}

gives

\frac{\pi}{6}

for

n=0

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196 Easy

196 of 261

Application +4 | -1

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

, then the value

\left( \mathrm{s} \right)

\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 solving the given trigonometric equation leads to

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

for specific values of n.

197 Easy

197 of 261

Analyzing +4 | -1

\cos \,\,\mathrm{\theta } = - \frac{1}{{\sqrt {2\;} }}

and

\tan \,\,\mathrm{\theta } = 1

, then the general value of

\mathrm{\theta }

View answer and explanation

Correct option: B

2\left( {n + 1} \right)\mathrm{\pi } + \frac{\mathrm{\pi }}{4}

The correct answer is B (

2(n+1)\pi + \frac{\pi}{4}

) because it represents the general solution that satisfies both the cosine and tangent conditions.

198 Easy

198 of 261

Application +4 | -1

\sin \,\,\mathrm{\theta } + \cos \,\,\mathrm{\theta } = 1

then the general value of

\mathrm{\theta }

View answer and explanation

Correct option: B

n\pi + {\left( { - 1} \right)^n}\frac{\pi }{4} - \frac{\pi }{4}

Option B (

n\pi + {\left( { - 1} \right)^n}\frac{\pi }{4} - \frac{\pi }{4}

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

199 Easy

199 of 261

Analyzing +4 | -1

The sum of solutions of the equation

\frac{{\cos x}}{{1 + \sin x}} = \left| {\tan 2x} \right|

x \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right) - \left\{ {\frac{\pi }{4}, - \left. {\frac{\pi }{4}} \right\}} \right.

View answer and explanation

Correct option: A

- \frac{{11\pi }}{{30}}

Option A (

- \frac{{11\pi }}{{30}}

) is correct because it is the sum of the solutions to the given equation within the specified interval.

200 Easy

200 of 261

Application +4 | -1

\cos \,\,p\;\theta = \cos \,\,q\;\theta ,\,\,p \ne q

then

View answer and explanation

Correct option: B

\theta = \frac{{2\;n\;\pi }}{{p \pm q}},\,\,n \in Z

Option B (

\theta = \frac{{2\;n\;\pi }}{{p \pm q}},\\,\,n \in Z

) is correct because it represents the general solution for the given trigonometric equation using the properties of cosine function.

201 Easy

201 of 261

Application +4 | -1

Total number of solutions of

\sin 2 x-\sin x-1=0

[-2 \pi, 2 \pi]

is equal to

View answer and explanation

Correct option: B

Option B (4) is correct. By simplifying the equation to

\sin x = \frac{{1 - \sqrt{{5}}}}{{2}}

, we find that there are 4 solutions for

x

in the interval

[-2\pi, 2\pi]

202 Easy

202 of 261

Application +4 | -1

Number of values of

x \in[0,4 \pi]

and satisfying

\sqrt{2} \sec x+\tan x=1

View answer and explanation

Correct option: C

The correct answer is C (2) because solving the trigonometric equation

\sqrt{2} \sec x + \tan x = 1

within the interval

[0, 4\pi]

yields 2 distinct solutions.

203 Easy

203 of 261

Application +4 | -1

If the distance between the points

\left( {a\cos \mathrm{\theta },a\sin \mathrm{\theta }} \right)

and

\left( {a\cos \phi ,a\sin \phi } \right)

2a

, then

\mathrm{\theta }

is equal to

View answer and explanation

Correct option: A

2n\pi \pm \pi + \phi ,n \in Z

Option A (

2n\pi \pm \pi + \phi ,n \in Z

) is correct because it represents the general solution for

\theta

derived from the given distance condition between the two points.

204 Easy

204 of 261

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.

205 Easy

205 of 261

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.

206 Easy

206 of 261

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}

207 Easy

207 of 261

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

208 Easy

208 of 261

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}

209 Easy

209 of 261

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.

210 Easy

210 of 261

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.

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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