The values of $\theta $ lying between $\theta = 0$ and $\theta = \frac{\pi }{2}$ and satisfying the equation $\left| {\begin{array}{*{20}{c}} {1 + {{\cos }^2}\theta }&{{{\sin }^2}\theta }&{4\sin 4\;\theta }\\ {{{\cos }^2}\theta }&{1 + {{\sin }^2}\theta }&{4\sin 4\;\theta }\\ {{{\cos }^2}\theta }&{{{\sin }^2}\theta }&{1 + 4\sin 4\;\theta } \end{array}} \right| = 0$, is

$\frac{{7\;\pi }}{{24}},\frac{{5\;\pi }}{{24}}$

$\frac{{5\,\pi }}{{24}},\frac{\pi }{{24}}$

The solution of the inequality ${\log _{1/2}}\sin x > {\rm{lo}}{{\rm{g}}_{1/2}}\cos x$ in $\left( {0,2\pi } \right)$ is

$x \in \left( {\frac{{5\pi }}{4},\,\;2\pi } \right)$

$x \in \left( {\;0,\,\,\frac{\pi }{4}} \right)\,\, \cup \,\,\left( {\frac{{5\pi }}{4},\;\,2\pi } \right)$

If $4\,\,\cos \,\,{\rm{\theta }} - 3\,\,\sec \,\,{\rm{\theta }} = 2\tan \,\,{\rm{\theta }},{\rm{\;then}}\,\,{\rm{\theta }}$ is equal to

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

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

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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 481-510 of 581 questions

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481 Easy

481 of 581

Analyzing +4 | -1

\cos x=\frac{2 \cos y-1}{2-\cos y}

, where

x, y \in(0, \pi)

, then

\tan \frac{x}{2} \cot \frac{y}{2}

is equal to

View answer and explanation

Correct option: B

\sqrt{3}

Option B (

\sqrt{3}

) is correct because by substituting the given expression for

\cos x

into the half-angle formula, we can simplify and find the value of

\tan \frac{x}{2} \cot \frac{y}{2}

482 Hard

482 of 581

Analyzing +4 | -1

Let

A, B

and

C

be three angles such that

A=\frac{\pi}{4}

and

\tan B \tan C=p

. Then, all possible values of

p

such that

A, B

and

C

are the angles of a triangle, is

View answer and explanation

Correct option: B

\geq(\sqrt{2}+1)^2

Option B (

\geq(\sqrt{2}+1)^2

) is correct because the condition for A, B, and C to be angles of a triangle leads to this inequality.

483 Medium

483 of 581

Application +4 | -1

\sin x+\sin ^2 x=1

then

\cos ^{12} x+3 \cos ^{10} x+3 \cos ^8 x+\cos ^6 x

is equal to -

View answer and explanation

Correct option: A

Option A (1) is correct because the expression simplifies to 1 after substituting

\cos^2 x = \sin x

484 Medium

484 of 581

Application +4 | -1

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

then

\cos (\theta-\pi / 4)

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is derived by using the trigonometric identity

\tan(\theta) = \cot(\frac{\pi}{2} - \theta)

and solving for

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

485 Medium

485 of 581

Application +4 | -1

\sin \theta=n \sin (\theta+2 \alpha)

, then

\tan (\theta+\alpha)

is equal to

View answer and explanation

Correct option: B

\left(\frac{1+n}{1-n}\right) \tan \alpha

Option B (

\left(\frac{1+n}{1-n}\right) \tan \alpha

) is correct because it is the result of applying the componendo and dividendo rule and trigonometric identities to the given equation.

486 Medium

486 of 581

Application +4 | -1

f(\theta)=(\sin \theta+\operatorname{cosec} \theta)^2+(\cos \theta+\sec \theta)^2

, then minimum value of

f(\theta)

View answer and explanation

Correct option: C

Option C (9) is correct because by simplifying the given expression and using the fact that

0 < \sin^2 2\theta \le 1

, the minimum value of

f(\theta)

is found to be 9.

487 Medium

487 of 581

Application +4 | -1

The values of

\theta

lying between

\theta = 0

and

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

and satisfying the equation

\left| {\begin{array}{cccccccccccccccccccc} {1 + {{\cos }^2}\theta }&{{{\sin }^2}\theta }&{4\sin 4\;\theta }\\ {{{\cos }^2}\theta }&{1 + {{\sin }^2}\theta }&{4\sin 4\;\theta }\\ {{{\cos }^2}\theta }&{{{\sin }^2}\theta }&{1 + 4\sin 4\;\theta } \end{array}} \right| = 0

, is

View answer and explanation

Correct option: A

\frac{{11\;\pi }}{{24}},\frac{{7\;\pi }}{{24}}

Option D (

\frac{\pi }{{24}},\frac{{11\;\pi }}{{24}}

) is correct because these are the solutions for

\theta

within the given range that satisfy the determinant equation.

488 Easy

488 of 581

Application +4 | -1

\tan \left( {\frac{{\alpha \,\pi }}{4}} \right) = \cot \left( {\frac{{\beta \,\pi }}{4}} \right)

, then

View answer and explanation

Correct option: D

\alpha + \beta = 2\left( {2n + 1} \right),\;\forall n

is an integer

Option D (

\alpha + \beta = 2\left( {2n + 1} \right),\\;\forall n

is an integer) is correct because by converting cotangent to tangent and solving the resulting trigonometric equation, we find that

\alpha + \beta

must be an even multiple of an odd number.

489 Medium

489 of 581

Analyzing +4 | -1

The critical points of the function

f\left( x \right) = 2{\sin ^2}\left( {\frac{x}{6}} \right) + \sin \left( {\frac{x}{3}} \right) - \left( {\frac{x}{3}} \right)

Whose coordinates satisfy the inequality

{x^2} - 10 < - 19.5x

View answer and explanation

Correct option: B

6\pi

Option B (

6\pi

) is correct because it is a critical point of the function that satisfies the given inequality.

490 Medium

490 of 581

Analyzing +4 | -1

\cos x-\sin \alpha \cot \beta \sin x=\cos \alpha

, then

\tan \frac{x}{2}

is equal to

View answer and explanation

Correct option: C

\tan \frac{\alpha}{2} \tan \frac{\beta}{2}

Option C (

\tan \frac{\alpha}{2} \tan \frac{\beta}{2}

) is correct because it is one of the solutions obtained after solving the trigonometric equation for

\tan \frac{x}{2}

491 Medium

491 of 581

Analyzing +4 | -1

The solution of the inequality

{\log _{1/2}}\sin x > \mathrm{lo}{\mathrm{g}_{1/2}}\cos x

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

View answer and explanation

Correct option: B

x \in \left( {0,\,\frac{\pi }{4}} \right)

Option B is correct because it finds the interval where both sin x > 0, cos x > 0, and sin x < cos x, which is (0, π/4).

492 Medium

492 of 581

Application +4 | -1

\sin x+\sin ^2 x=1

, then the value of

\cos ^2 x+\cos ^4 x+\cot ^4 x-\cot ^2 x

is equal to

View answer and explanation

Correct option: C

Option C (2) is correct because by using the given equation and trigonometric identities, the expression simplifies to 2.

493 Easy

493 of 581

Application +4 | -1

The general value of

\theta

in the equation

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

View answer and explanation

Correct option: B

2n\pi \,\, \pm \,\,\frac{{7\pi }}{4},\;\,n \in I

Option B (

2n\pi \pm \frac{{7\pi }}{{4}},\;n \in I

) is correct because it represents the general solution for

\theta

that satisfies both

\cos \theta = \frac{1}{\sqrt{2}}

and

\tan \theta = -1

494 Easy

494 of 581

Application +4 | -1

The general solution of

\sin \,\,x - \cos \,\,x = \sqrt 2

, for any integer

n

View answer and explanation

Correct option: B

2n\pi + \frac{{3\pi }}{4}

Option B (

2n\pi + \frac{{3\pi }}{{4}}

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

\sin x - \cos x = \sqrt{2}

after simplification and solving for x.

495 Medium

495 of 581

Analyzing +4 | -1

Total number of solutions of

\tan x+\cot x=2 \operatorname{cosec} x

[-2 \pi, 2 \pi]

View answer and explanation

Correct option: C

Option B (

\frac{{3 \pi}}{{2}}

) is correct because the equation has at least one solution only if

\sin(ax+b) = -1

when

x=1

, leading to

a+b = \frac{{3\pi}}{{2}}

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496 Easy

496 of 581

Analyzing +4 | -1

4\,\,\cos \,\,\mathrm{\theta } - 3\,\,\sec \,\,\mathrm{\theta } = 2\tan \,\,\mathrm{\theta },\mathrm{\;then}\,\,\mathrm{\theta }

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it represents the general solution of the given trigonometric equation after simplification and solving for

\theta

497 Easy

497 of 581

Analyzing +4 | -1

u=\sqrt{a^{2} \cos ^{2} \theta+b^{2} \sin ^{2} \theta}+\sqrt{a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta}

then the difference between the maximum and minimum values of

u^{2}

is given by

View answer and explanation

Correct option: A

(a-b)^{2}

Option A ((a-b)²) is correct because after simplifying the expression for u² and finding its maximum and minimum values, the difference between them is (a-b)².

498 Easy

498 of 581

Analyzing +4 | -1

\frac{1+\sin 2 x}{1-\sin 2 x}=\cot ^2(a+x)

\forall \, x \in R \sim\left(n \pi+\frac{\pi}{4}\right), n \in N

, then

a

is equal to

View answer and explanation

Correct option: C

\frac{3 \pi}{4}

Option C (

\frac{3 \pi}{4}

) is correct because by simplifying the given expression and using trigonometric identities, we can find that

a = \frac{3 \pi}{4}

499 Easy

499 of 581

Application +4 | -1

\tan 9 \theta=\frac{3}{4}

where

\left(0<\theta<\frac{\pi}{18}\right)

Then value of

3 \operatorname{cosec} 3 \theta-4 \sec 3 \theta=

View answer and explanation

Correct option: D

Option D (10) is correct because by using trigonometric identities and the given information, the expression simplifies to 10.

500 Medium

500 of 581

Application +4 | -1

The solution of the equation

{\left[ {\sin \,\,x + \cos \,\,x} \right]^{1 + \sin 2x}} = 2

- \,\pi \le x \le \pi

View answer and explanation

Correct option: C

\frac{\pi }{4}

Option C (π/4) is correct because substituting x = π/4 into the equation satisfies the given condition.

501 Easy

501 of 581

Application +4 | -1

0 \leq \mathrm{x} \leq \pi

, then the solution of the equation

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

is given by

x

equal to-

View answer and explanation

Correct option: A

\pi / 6, \pi / 3

Option A (

\pi / 6, \pi / 3

) is correct because these are the solutions to the trigonometric equation derived from the given exponential equation within the specified range.

502 Medium

502 of 581

Application +4 | -1

If the equation

\sin x+\cos (k+x)+\cos (k-x)=2

has real solutions, then the complete set of values of

k

(n \in I)

View answer and explanation

Correct option: A

\left[\mathrm{n} \pi-\frac{\pi}{6}, \mathrm{n} \pi+\frac{\pi}{6}\right]

Option A ([nπ - π/6, nπ + π/6]) is correct because the given equation has real solutions only when cos²(k) >= 3/4, which leads to the specified interval for k.

503 Medium

503 of 581

Analyzing +4 | -1

\cot ^2 x=\cot (x-y) \cot (x-z)

, then

\cot 2 x

is equal to (where,

x \neq \pm \pi / 4

)

View answer and explanation

Correct option: B

\frac{1}{2}(\cot y+\cot z)

Option B (

\frac{1}{2}(\cot y+\cot z)

) is correct because after expanding and simplifying the given equation,

\cot 2x

can be expressed in terms of

\cot y

and

\cot z

504 Hard

504 of 581

Understanding +4 | -1

The number of all possible

5-

tuples

\left(a_{1}, a_{2}, a_{3}\right.

a_{4}, a_{5}

) such that

a_{1}+a_{2} \sin x+a_{3} \cos x+a_{4} \sin 2 x+a_{5}

\cos 2 x=0

holds for all

x

View answer and explanation

Correct option: B

Option B (1) is correct because the given equation holding for all

x

implies that all coefficients must be zero, leading to only one possible 5-tuple (0, 0, 0, 0, 0).

505 Easy

505 of 581

Analyzing +4 | -1

The set of values of

x

for which

\sin x \cos ^3 x>\cos x \sin ^3 x, 0 \leq x \leq 2 \pi

, is

View answer and explanation

Correct option: B

\left(0, \frac{\pi}{4}\right)

Option B (

\left(0, \frac{\pi}{4}\right)

) is correct because it represents the interval where the inequality

\sin x \cos ^3 x > \cos x \sin ^3 x

holds true for

0 \leq x \leq 2 \pi

506 Medium

506 of 581

Analyzing +4 | -1

For all

\theta

\left[0, \frac{\pi}{2}\right], \cos (\sin \theta)

View answer and explanation

Correct option: B

> \sin (\cos \theta)

Option B (

> \sin (\cos \theta)

) is correct because

\cos(\sin \theta) > \sin(\cos \theta)

for all

\theta

[0, \frac{\pi}{2}]

507 Medium

507 of 581

Analyzing +4 | -1

A, B

and

C

are the angles of a triangle such that

\sec (A-B), \sec A

and

\sec (A+B)

are in AP, then

View answer and explanation

Correct option: B

2 \sec ^2 A=\sec ^2 \frac{B}{2}

Option B (

2 \sec^2 A = \sec^2 \frac{{B}}{{2}}

) is correct because using the arithmetic progression condition and trigonometric identities leads to this relationship.

508 Easy

508 of 581

Application +4 | -1

Let

S

be the set of all

\alpha \in R

such that the equation,

\cos 2x + \alpha \sin x = 2\alpha - 7

has a solution. Then,

S

is equal to

View answer and explanation

Correct option: D

[2, 6]

Option D ([2, 6]) is correct because it represents the range of values for

\alpha

for which the given equation has a solution, derived by analyzing the range of the sine function.

509 Medium

509 of 581

Application +4 | -1

D

is a point on the side

B C

of a

\triangle A B C

such that

B D: D C=m: n

and

\angle A D C=\theta

\angle B A D=\alpha

and

\angle D A C=\beta

. Then,

(m+n) \cot \theta

is equal to

View answer and explanation

Correct option: C

m \cot \alpha-n \cot \beta

Option C (

m \cot \alpha - n \cot \beta

) is correct because it is the result of applying trigonometric identities and the sine rule to the given triangle configuration.

510 Medium

510 of 581

Analyzing +4 | -1

Which one of the following equations has no solution ?

View answer and explanation

Correct option: B

\mathrm{cosec}\;\,\theta .\sec \,\,\theta = 1

The correct answer is B (

\mathrm{cosec}\;\,\theta .\sec \,\,\theta = 1

) because it leads to sin θ cos θ = 1, which implies sin 2θ = 2, which is impossible since the range of sine function is [-1, 1].

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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