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

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

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

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

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

The set of all $x$ in $(-\pi, \pi)$ satisfying $|4 \sin x-1| < \sqrt{5}$ is given by

$x \in\left(-\frac{\pi}{10}, \pi\right)$

$x \in\left(-\pi, \frac{3 \pi}{10}\right)$

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

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

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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 421-450 of 581 questions

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421 Hard

421 of 581

Analyzing +4 | -1

In any

\Delta \,ABC,\,\;\mathrm{II}\,\,\left( {\frac{{{{\sin }^2}\,\,A + \sin \,\,A + 1}}{{\sin \,\,A}}} \right)

is always greater than

View answer and explanation

Correct option: C

Option C (27) is correct because the minimum value of the expression is 27, found by applying AM-GM inequality.

422 Easy

422 of 581

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.

423 Easy

423 of 581

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

424 Easy

424 of 581

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.

425 Easy

425 of 581

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.

426 Medium

426 of 581

Analyzing +4 | -1

The number of solution of

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

View answer and explanation

Correct option: C

Zero solution

Option C (Zero solution) is correct because the left-hand side is bounded by 2, while the right-hand side is always greater than or equal to 6.

427 Easy

427 of 581

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.

428 Medium

428 of 581

Analyzing +4 | -1

f(\theta)=5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3

, then range of

f(\theta)

View answer and explanation

Correct option: D

[-4,10]

Option D ([-4,10]) is correct because it represents the range of the given trigonometric function after simplification and applying the range formula.

429 Medium

429 of 581

Application +4 | -1

The equation

\left( {\cos \,p - 1} \right){x^2} + \left( {\cos \,p} \right)x + \sin p = 0

, where

x

is a variable, has real roots. Then, the interval of

p

may be any one of the following:

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because the condition for real roots of the quadratic equation is satisfied within this interval.

430 Medium

430 of 581

Analyzing +4 | -1

\exp \left\{\left(\sin ^2 x+\sin ^4 x+\sin ^6 x+\ldots\right.\right.

upto

\left.\left.\infty\right) \ln 2\right\}

satisfies the equation

x^2-17 x+16=0

then value of

\frac{2 \cos x}{\sin x+2 \cos x}(0 < x < \pi / 2)

View answer and explanation

Correct option: A

\frac{1}{2}

The correct answer is

\frac{1}{2}

because solving the quadratic equation gives

\tan^2 x = 4

, leading to

\tan x = 2

, and substituting this into the expression yields

\frac{1}{2}

431 Easy

431 of 581

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

432 Hard

432 of 581

Analyzing +4 | -1

Number of integral solutions of the equation

\sin ^{-1}|\cos x|=\cos ^{-1}|\sin x|

[0,4 \pi]

is equal to

View answer and explanation

Correct option: D

The correct answer is 13 because the equation

\sin^{-1}|\cos x| = \cos^{-1}|\sin x|

holds for all real numbers, and there are 13 integers in the interval [0, 4

\pi

] that satisfy the equation after simplification.

433 Medium

433 of 581

Application +4 | -1

\sin \,\,5\;x + \sin \,\,3\;x + \sin \,\,x = 0

then the value of

x

other than zero, lying between

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

View answer and explanation

Correct option: C

\frac{\pi }{3}

Option C is correct because the value of x is found by solving the trigonometric equation using sum-to-product formulas.

434 Medium

434 of 581

Application +4 | -1

The general solution for

\alpha

is, if

\sin \alpha, 1, \cos 2 \alpha

in G.P.

View answer and explanation

Correct option: A

n \pi+(-1)^{n+1} \frac{\pi}{2}

Option A (

n \pi+(-1)^{{n+1}} \frac{{\pi}}{{2}}

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

\alpha

435 Easy

435 of 581

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.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

436 Easy

436 of 581

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.

437 Easy

437 of 581

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.

438 Medium

438 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}

Option A (

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

) is correct because solving the equation

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

leads to

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

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

, and considering the given range,

x = \frac{{\pi}}{{6}}

is a valid solution.

439 Medium

439 of 581

Application +4 | -1

If the complex numbers

\sin \,\,x + \,\,i\,\,\cos \,\,2\;x

and

\cos \,\,x - i\,\,\sin \,\,2\;x

are conjugate to each other, then

x

is equal to

View answer and explanation

Correct option: D

None of these

Option D is correct because no value of x satisfies the condition that the given complex numbers are conjugates of each other.

440 Medium

440 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

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

Option B (

x \in(-\frac{\pi}{10}, \frac{3 \pi}{10})

) is correct because solving the inequality

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

within the interval

(-\pi, \pi)

leads to this solution set.

441 Easy

441 of 581

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.

442 Medium

442 of 581

Application +4 | -1

The expression

\left(\frac{\cos \mathrm{A}+\cos \mathrm{B}}{\sin \mathrm{A}-\sin \mathrm{B}}\right)^{\mathrm{n}}+\left(\frac{\sin \mathrm{A}+\sin \mathrm{B}}{\cos \mathrm{A}-\cos \mathrm{B}}\right)^{\mathrm{n}}=

View answer and explanation

Correct option: A

2 \cot ^{\mathrm{n}}\left(\frac{\mathrm{A}-\mathrm{B}}{2}\right)

\mathrm{n}

is even

Option A is correct because the expression simplifies to a trigonometric form, and the result depends on whether n is even or odd.

443 Easy

443 of 581

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.

444 Medium

444 of 581

Understanding +4 | -1

\sin \frac{\pi}{n}+\sin \frac{3 \pi}{n}+\sin \frac{5 \pi}{n}+\ldots

n

terms is equal to

View answer and explanation

Correct option: D

Option D is correct because the sum of the sine series is 0.

445 Medium

445 of 581

Application +4 | -1

Solve the equations for

x

and

y:

\begin{aligned} & x \cos ^3 y+3 x \cos y \sin ^2 y=14 \\ & x \sin ^3 y+3 x \cos ^2 y \sin y=13 \end{aligned}

View answer and explanation

Correct option: C

Both

Option C (Both) is correct because by dividing and using componendo and dividendo, we find two possible solutions for x and y that satisfy the given conditions.

446 Easy

446 of 581

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]

447 Medium

447 of 581

Analyzing +4 | -1

f(x)=\left|\begin{array}{ccc}\sin ^2 \theta & \cos ^2 \theta & x \\ \cos ^2 \theta & x & \sin ^2 \theta \\ x & \sin ^2 \theta & \cos ^2 \theta\end{array}\right|, \theta \in(0, \pi / 2)

, then roots of

f(x)=0

are

View answer and explanation

Correct option: B

1 / 2,-1

Option B (

1/2, -1

) is the correct answer because these are the values of x that make the determinant of the given matrix equal to zero.

448 Easy

448 of 581

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.

449 Easy

449 of 581

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.

450 Medium

450 of 581

Application +4 | -1

The value of

\theta

satisfying

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

View answer and explanation

Correct option: B

60^{\circ}

Option B (

60^{\circ}

) is correct because by rewriting the equation and using the cosine subtraction formula, we find that

\theta = 60^{\circ}

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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