If $\sin x+\sqrt{3} \cos x=\sqrt{2}$, then $x$ is

$2 n \pi+\frac{\pi}{12}, 2 n \pi-\frac{\pi}{3}$

$2 n \pi+\frac{\pi}{12}, 2 n \pi-\frac{\pi}{12}$

If $\alpha+\beta+\gamma=2 \pi$, then

$\tan \frac{\alpha}{2} \tan \frac{\beta}{2}+\tan \frac{\beta}{2} \tan \frac{\gamma}{2}+\tan \frac{\gamma}{2} \tan \frac{\alpha}{2}=1$

If $\sin \theta=\sin \alpha$, then

$\frac{\theta+\alpha}{2}$ is any odd multiple of $\pi$ and $\frac{\theta-\alpha}{2}$ is any multiple of $\pi$

$\frac{\theta+\alpha}{2}$ is any multiple of $\frac{\pi}{2}$ and $\frac{\theta-\alpha}{2}$ is any even multiple of $\pi$

$\frac{\theta+\alpha}{2}$ is any even multiple of $\frac{\pi}{2}$ and $\frac{\theta-\alpha}{2}$ is any odd multiple of $\pi$

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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Medium

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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 271-294 of 294 questions

271 Medium

271 of 294

Analyzing +4 | -1

x=\alpha, \beta

satisfies both the equation

\cos ^2 x+a \cos x+b=0

and

\sin ^2 x+p \sin x+q=0

, then the relation between

a, b, p

and

q

View answer and explanation

Correct option: C

2(b+q)=a^2+p^2-2

Option C (

2(b+q)=a^2+p^2-2

) is correct because it represents the relationship between the coefficients derived from the given equations and trigonometric identities.

272 Medium

272 of 294

Analyzing +4 | -1

The equation

\tan ^4 x-2 \sec ^2 x+a^2=0

will have atleast one solution, if

View answer and explanation

Correct option: C

|a| \leq \sqrt{3}

Option C (

|a| \leq \sqrt{3}

) is correct because the equation has at least one solution when the discriminant of the transformed equation is non-negative.

273 Medium

273 of 294

Analyzing +4 | -1

In a

\triangle A B C

, the maximum value of

\frac{a \cos ^2 \frac{A}{2}+b \cos ^2 \frac{B}{2}+c \cos ^2 \frac{C}{2}}{a+b+c}

View answer and explanation

Correct option: C

\frac{3}{4}

Option C (

\frac{3}{4}

) is correct because the maximum value of the expression is

\frac{3}{4}

when using trigonometric identities and the inequality

R \geq 2r

274 Medium

274 of 294

Analyzing +4 | -1

The sum of the series

\sum\limits_{n = 1}^\infty {\sin \left( {\frac{{n!\,\,\pi }}{{720}}} \right)}

View answer and explanation

Correct option: C

\sin \left(\frac{\pi}{6}\right)+\sin \left(\frac{\pi}{30}\right)+\sin \left(\frac{\pi}{120}\right)+\sin \left(\frac{\pi}{360}\right)+\sin \left(\frac{\pi}{720}\right)

Option C (

\sin \left(\frac{\pi}{6}\right)+\sin \left(\frac{\pi}{30}\right)+\sin \left(\frac{\pi}{120}\right)+\sin \left(\frac{\pi}{360}\right)+\sin \left(\frac{\pi}{720}\right)

) is correct because after the first five terms, all subsequent terms become multiples of

\pi

and thus their sine values are zero.

275 Medium

275 of 294

Application +4 | -1

\sin x+\sqrt{3} \cos x=\sqrt{2}

, then

x

View answer and explanation

Correct option: B

2 n \pi+\frac{5 \pi}{12}, 2 n \pi-\frac{\pi}{12}

Option B (

2 n \pi+\frac{{5 \pi}}{{12}}, 2 n \pi-\frac{{\pi}}{{12}}

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

276 Medium

276 of 294

Analyzing +4 | -1

Consider the following statements :

1.

\cot \,\,\mathrm{\theta } - \tan \,\,\mathrm{\theta },\,\,\;\mathrm{then}\,\,\mathrm{\;\theta } = \left( {4n + 1} \right)\,\,\frac{\pi }{8}

\sin \,\,2x\,\, + \,\,\cos \,\,2x\,\, + \,\,\sin \,\,x + \cos \,\,x + 1 = 0

has no solution in the Ist quadrant.
Which of these is/are correct?

View answer and explanation

Correct option: C

Both of these

Option C (Both of these) is correct because both statements regarding trigonometric identities and solutions are accurate.

277 Medium

277 of 294

Application +4 | -1

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

, then

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

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the value obtained after simplifying the given trigonometric equation and using the cosine subtraction formula.

278 Medium

278 of 294

Application +4 | -1

The value of

\cos \frac{\pi}{7} \cos \frac{4 \pi}{7} \cos \frac{5 \pi}{7}

View answer and explanation

Correct option: D

\frac{1}{8}

The correct answer is

\frac{1}{8}

because the expression simplifies using trigonometric identities and the double angle formula.

279 Medium

279 of 294

Application +4 | -1

\sin (\alpha+\beta)=1

and

\sin (\alpha-\beta)=\frac{1}{2}

, then

\tan (\alpha+2 \beta) \tan (2 \alpha+\beta)

is equal to

View answer and explanation

Correct option: A

Option A (1) is correct because solving the system of equations for

\alpha

and

\beta

and substituting into the expression yields a value of 1.

280 Medium

280 of 294

Application +4 | -1

The value of

\cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{16 \pi}{15}

View answer and explanation

Correct option: A

\frac{1}{16}

Option A (

\frac{{1}}{{16}}

) is correct because the given expression simplifies to

\frac{{1}}{{16}}

by repeatedly using the identity

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

281 Medium

281 of 294

Application +4 | -1

\alpha+\beta+\gamma=2 \pi

, then

View answer and explanation

Correct option: A

\tan \frac{\alpha}{2}+\tan \frac{\beta}{2}+\tan \frac{\gamma}{2}=\tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2}

Option A (

\tan \frac{\alpha}{2}+\tan \frac{\beta}{2}+\tan \frac{\gamma}{2}=\tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2}

) is correct because it is derived from the given condition

\alpha+\beta+\gamma=2 \pi

by using the tangent addition formula.

282 Medium

282 of 294

Analyzing +4 | -1

\cos \,\,x \ne - \frac{1}{2}

, then the solutions of

\cos \,\,x + \cos \,\,2x + \cos \,\,3x = 0

are

View answer and explanation

Correct option: A

2n\pi \,\, \pm \,\,\frac{\pi }{4},\,n \in Z

Option A (

2n\pi \, \pm \, \frac{\pi }{4},\,n \in Z

) is correct because solving the trigonometric equation

\cos x + \cos 2x + \cos 3x = 0

yields this general solution.

283 Medium

283 of 294

Application +4 | -1

Let

\cos (\alpha+\beta)=\frac{4}{5}

and

\sin (\alpha-\beta)=\frac{5}{13}

, where

0 \leq \alpha, \beta \leq \frac{\pi}{4}

. Then,

\tan 2 \alpha

is equal to

View answer and explanation

Correct option: B

\frac{56}{33}

Option B (

\frac{56}{33}

) is correct because it is the value of

\tan 2\alpha

calculated using the given trigonometric values and the tangent addition formula.

284 Medium

284 of 294

Application +4 | -1

The value of

\tan ^{2} \theta \sec ^{2} \theta\left(\cot ^{2} \theta-\cos ^{2} \theta\right)

View answer and explanation

Correct option: B

Option B is correct because simplifying the given expression using trigonometric identities leads to the value 1.

285 Medium

285 of 294

Analyzing +4 | -1

The equation

{\cos ^4}x - \left( {\lambda + 2} \right){\cos ^2}x - \left( {\lambda + 3} \right) = 0

possesses a solution, if

View answer and explanation

Correct option: C

- 3 \le \lambda \le - 2

Option C (

- 3 \le \lambda \le - 2

) is correct because it is the range of

\lambda

that satisfies the condition

0 \le {\cos ^2}x\, \le 1

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286 Medium

286 of 294

Application +4 | -1

The complete solution of the equation

7 \cos ^2 x+\sin x \cos x-3=0

is given by:

View answer and explanation

Correct option: D

{n} \pi+\frac{3 \pi}{4}, {k} \pi+\tan -1\left(\frac{4}{3}\right)({k}, {n} \in {Z})

Option D (

{n} \pi+\frac{{3 \pi}}{{4}}, {k} \pi+\tan -1(\frac{{4}}{{3}})({k}, {n} \in {Z})

) is correct because it represents the complete solution set obtained by solving the trigonometric equation after converting it to a quadratic in tan x.

287 Medium

287 of 294

Application +4 | -1

Solution of the inequality

\tan \left( {x + \frac{\pi }{3}} \right) \ge 1\;

View answer and explanation

Correct option: B

\left( {n\pi - \frac{\pi }{{12}},n\pi + \frac{\pi }{6}} \right)

Option B is correct because it represents the solution set for the given inequality, derived by considering the range of the tangent function.

288 Medium

288 of 294

Analyzing +4 | -1

The values of

\theta

, for which the system

(\sin 3 \theta) x-y+z=0,(\cos 2 \theta) x+4 y+3 z=0

and

2 x+7 y+7 z=0

, has a non-trivial solution, are

View answer and explanation

Correct option: A

n \pi, n \pi+(-1)^n \frac{\pi}{6}

Option A (

n \pi, n \pi+(-1)^n \frac{\pi}{6}

) is correct because these are the values of θ that make the determinant of the coefficient matrix zero, leading to non-trivial solutions.

289 Medium

289 of 294

Application +4 | -1

|\tan x| \leq 1

and

x \in[-\pi, \pi]

, then the solution set for

x

is:

View answer and explanation

Correct option: A

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

Option A is correct because it represents the intervals where the absolute value of the tangent function is less than or equal to 1 within the given range of

[-\pi, \pi]

290 Medium

290 of 294

Application +4 | -1

If the equation

4 \cos ^2 x \sin x-2 \sin ^2 x=3 \sin x

, then

x

is equal to-

View answer and explanation

Correct option: B

n \pi+(-1)^{n+1}\left(\frac{3 \pi}{10}\right)

Option B (

n \pi+(-1)^{{n+1}}(\frac{{3 \pi}}{{10}})

) is correct because it represents the general solution of the given trigonometric equation, derived by factoring and solving for x.

291 Medium

291 of 294

Analyzing +4 | -1

\sin \theta=\sin \alpha

, then

View answer and explanation

Correct option: A

\frac{\theta+\alpha}{2}

is any odd multiple of

\frac{\pi}{2}

and

\frac{\theta-\alpha}{2}

is any multiple of

\pi

Option A (

\frac{\theta+\alpha}{2}

is any odd multiple of

\frac{\pi}{2}

and

\frac{\theta-\alpha}{2}

is any multiple of

\pi

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

292 Medium

292 of 294

Application +4 | -1

\cos x \cos 2 x \cos 4 x \cos 8 x=\frac{\operatorname{cosec} x}{16}

then-

View answer and explanation

Correct option: A

x=(4 n+1) \frac{\pi}{32}

Option A (

x=(4 n+1) \frac{\pi}{32}

) is correct because it represents the solution obtained after repeatedly applying the identity 2sin(x)cos(x) = sin(2x) and solving the resulting equation.

293 Medium

293 of 294

Application +4 | -1

The general solution of the equation

\tan 2\theta \,.\,\,\tan \theta = 1

for

n \in Z

View answer and explanation

Correct option: B

\left( {2n + 1} \right)\frac{\pi }{6}

Option B (

\left( {2n + 1} \right)\frac{\pi }{6}

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

294 Medium

294 of 294

Application +4 | -1

\tan \alpha=\frac{1}{7}

and

\sin \beta=\frac{1}{\sqrt{10}}

, where

0 < \alpha, \beta < \frac{\pi}{2}

, then

2 \beta

is equal to

View answer and explanation

Correct option: A

\frac{\pi}{4}-\alpha

Option A (

\frac{{\pi}}{{4}} - \alpha

) is correct because by calculating

\sin 2\beta

and then

\tan(\alpha + 2\beta)

, we find that

\alpha + 2\beta = \frac{{\pi}}{{4}}

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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