If $0 < x < \pi $ and $\cos \,\,x + \sin \,\,x = \frac{1}{2}$, then $\tan \,\,x$ is

$\frac{{\left( {1 + \sqrt 7 } \right)}}{4}$

$\frac{{\left( {1 - \sqrt 7 } \right)}}{4}$

Solution set of the inequality $|\tan x| < 1 / \sqrt{3}$ is

$\left(n \pi-\frac{\pi}{3}, n \pi+\frac{\pi}{3}\right) ; n \in Z$

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

The solution set of the inequation ${\log _{1/2}}\sin x > {\log _{1/2}}\cos x$ in $\left[ {0,\;\,2\pi } \right]$, is

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

$\left( { - \pi /4,\,\pi /4} \right)$

Solution for $x$ and $y$ when $x^2+2 x \sin x y+1=0$ is

$x=1, y=2 n \pi, \mathrm{n} \in \mathbb{Z}$

$x=-1, y=(2 n+1) \pi, \mathrm{n} \in \mathbb{Z}$

The solution set of inequality $\cos x \geq-\frac{1}{2}$ is

$\bigcup_\limits {n \in Z}\left(n \pi-\frac{2 \pi}{3}, n \pi+\frac{2 \pi}{3}\right)$

$\bigcup_\limits {n \in Z}\left(2 n \pi-\frac{2 \pi}{3}, n \pi-\frac{\pi}{3}\right)$

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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Hard

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Practice format and chapter quiz

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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1 Hard

1 of 26

Analyzing +4 | -1

The least positive non-integral solution of

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

, is

View answer and explanation

Correct option: C

Irrational of the form

\frac{{\sqrt p - 1}}{4}

, when

\frac{{\sqrt p + 1}}{4}

is an odd integer

Option C is correct because the solution to the given trigonometric equation results in an irrational number of the form (

\sqrt{p}

- 1)/4, where (

\sqrt{p}

+ 1)/4 is an odd integer.

2 Hard

2 of 26

Application +4 | -1

Solution of the equation

\sin x+\cos x-2 \sqrt{2} \sin x \cos x=0

View answer and explanation

Correct option: A

x=(8 k+3) \cdot \frac{\pi}{12}

x=2 k \pi+\frac{\pi}{4} ; k \in \mathbb{Z}

Option A is the correct solution as it represents the general solutions derived from solving the trigonometric equation by converting it into

\sin(x + \frac{\pi}{4}) = \sin(2x)

3 Hard

3 of 26

Analyzing +4 | -1

The value of

\sum\limits_{{n=1}}^{\infty} \frac{\tan \left(\frac{\theta}{2^n}\right)}{2^{n-1} \cos \left(\frac{\theta}{2^{n-1}}\right)}

View answer and explanation

Correct option: A

\frac{2}{\sin 2 \theta}-\frac{1}{\theta}

Option A (

\frac{{2}}{{\sin 2 \theta}} - \frac{{1}}{{\theta}}

) is correct because the infinite sum telescopes and simplifies to this expression.

4 Hard

4 of 26

Analyzing +4 | -1

If the expression

\frac{{\sin \,\,\frac{x}{2}\,\, + \,\,\cos \,\,\frac{x}{2} - \,\,i\,\,\tan \,\,x}}{{1 + 2\,i\,\,\sin \,\,\frac{x}{2}}}

is real, then x is equal to

View answer and explanation

Correct option: C

2n\;\pi + 2{\tan ^{ - 1}}k

, where

k \in \left( {1,2} \right),\,\,n \in Z

Option C (

2n\\;\pi + 2{\tan ^{ - 1}}k

, where

k \in \left( {1,2} \right),\\,n \in Z

) is correct because it represents the solution for x when the given expression is real, derived by setting the imaginary part to zero.

5 Hard

5 of 26

Application +4 | -1

|k|=5

and

0^{\circ} \leq \theta \leq 360^{\circ}

, then the number of different solution of

3 \cos \theta+4 \sin \theta=k

View answer and explanation

Correct option: B

Two

Option B (Two) is correct because the equation has two solutions when

|k| = 5

and

0^{\circ} \leq \theta \leq 360^{\circ}

6 Hard

6 of 26

Application +4 | -1

The general solution for

\alpha

is, if

2 \cos ^2 \theta+3 \sin \theta=0

View answer and explanation

Correct option: C

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

Option A (

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

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

7 Hard

7 of 26

Analyzing +4 | -1

\left( {{{\tan }^2}\frac{\pi }{7} + {{\tan }^2}\frac{{2\pi }}{7} + {{\tan }^2}\frac{{3\pi }}{7}} \right)\left( {{{\cot }^2}\frac{\pi }{7} + {{\cot }^2}\frac{{2\pi }}{7} + {{\cot }^2}\frac{{3\pi }}{7}} \right)

is equal to

View answer and explanation

Correct option: B

105

Option B (105) is correct because by using trigonometric identities and Vieta's formulas, the expression simplifies to 105.

8 Hard

8 of 26

Application +4 | -1

The ordered pair

(x, y)

satisfying

1-2 x-x^2=\tan ^2(x+y)+\cot ^2(x+y)

View answer and explanation

Correct option: C

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

Option C ((-1,(2n+1)

\frac{π}{4}

+1)) is correct because the equation holds true only when both sides are equal to 2, which occurs when x = -1 and y = (2n+1)

\frac{π}{4}

+1.

9 Hard

9 of 26

Analyzing +4 | -1

\cos \,\,2x + k\,sin\,\,\;x = 2k - 7

has a solution for

View answer and explanation

Correct option: A

2 \le k \le 6

Option A (

2 \le k \le 6

) is correct because this range of

k

ensures that the value of

\sin x

obtained from the equation lies between -1 and 1, which is a necessary condition for a solution to exist.

10 Hard

10 of 26

Analyzing +4 | -1

If the equation

\cos \,\,\left( {\lambda \,\,\sin \,\,\theta } \right) = \sin \left( {\lambda \,\,\cos \,\,\theta } \right)

has a solution in

\left[ {0,\;\,2\;\pi } \right]

, then the smallest positive value of

\lambda

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it represents the smallest positive value of

\lambda

that allows the given trigonometric equation to have a solution within the specified interval.

11 Hard

11 of 26

Analyzing +4 | -1

Equation

\cos \,\,2x + 7 = a\left( {2 - \sin \,x} \right)

can have a real solution for

View answer and explanation

Correct option: B

a \in \left[ {2,\;\,6} \right]

Option B (

a \in [2, 6]

) is correct because the equation has a real solution only when the value of 'a' falls within this interval, ensuring that

\sin x

remains within its valid range [-1, 1].

12 Hard

12 of 26

Application +4 | -1

0 < x < \pi

and

\cos \,\,x + \sin \,\,x = \frac{1}{2}

, then

\tan \,\,x

View answer and explanation

Correct option: B

- \frac{{\left( {4 + \sqrt 7 } \right)}}{3}

Option B (

- \frac{{\left( {4 + \sqrt 7 } \right)}}{3}

) is correct because solving the given equation for

\tan x

within the specified interval yields this value.

13 Hard

13 of 26

Application +4 | -1

Solution set of the inequality

|\tan x| < 1 / \sqrt{3}

View answer and explanation

Correct option: A

\left(n \pi-\frac{\pi}{6}, n \pi+\frac{\pi}{6}\right) ; n \in Z

Option A ((

n \pi-\frac{\pi}{6}, n \pi+\frac{\pi}{6}

);

n \in Z

) is correct because it represents the general solution for the inequality

|\tan x| < 1 / \sqrt{3}

14 Hard

14 of 26

Understanding +4 | -1

Consider the following statements:

I. The function

f(x)=

greatest integer

\leq x, x \in R

is a continuous function.

II. All trigonometric functions are continuous on

R

. Which of the statements given above is/are correct?

View answer and explanation

Correct option: D

Neither I nor II

Option D is correct because neither the greatest integer function nor all trigonometric functions are continuous on R.

15 Hard

15 of 26

Application +4 | -1

The general solution of

{e^{ - 1/\sqrt 2 }}\,\,\left( {{e^{\sin x}} + \,\,{e^{\cos x}}} \right) = 2

View answer and explanation

Correct option: B

x = \frac{{\left( {4m + 1} \right)\pi }}{4}

Option B is correct because it satisfies the given equation by ensuring that both

\sin x

and

\cos x

are equal, leading to the minimum value of the expression.

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16 Hard

16 of 26

Analyzing +4 | -1

The solution set of the inequation

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

\left[ {0,\;\,2\pi } \right]

, is

View answer and explanation

Correct option: C

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

Option C is correct because it correctly identifies the interval where sin x < cos x and both sin x and cos x are positive within the given range [0, 2π].

17 Hard

17 of 26

Analyzing +4 | -1

The number of solutions of the equation

{x^3} + {x^2} + 4x + 2\,\,\sin \,\,x = 0\,\,\mathrm{in}\,\;0 \le \,\,x\,\, \le \,\,2\pi

View answer and explanation

Correct option: B

One

Option B (One) is correct because x=0 is the only solution that satisfies the given equation in the specified interval.

18 Hard

18 of 26

Application +4 | -1

Solution for

x

and

y

when

x^2+2 x \sin x y+1=0

View answer and explanation

Correct option: A

x= \pm 1, y=2 n \pi-\pi / 2 ; \mathrm{n} \in \mathbb{Z}

Option A (

x= \pm 1, y=2 n \pi-\pi / 2 ; \mathrm{{n}} \in \mathbb{{Z}}

) is correct. The equation is satisfied only when

x = \pm 1

and

\sin(xy) = -1

, leading to the given solution.

19 Hard

19 of 26

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.

20 Hard

20 of 26

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.

21 Hard

21 of 26

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.

22 Hard

22 of 26

Application +4 | -1

\theta

lies in the first quadrant and

\cos \theta

=8 / 17

, then the value of

\cos \left(30^{\circ}+\theta\right)+\cos \left(45^{\circ}-\theta\right)+

\cos \left(120^{\circ}-\theta\right)

View answer and explanation

Correct option: A

\left(\frac{\sqrt{3}-1}{2}+\frac{1}{\sqrt{2}}\right) \frac{23}{17}

Option A is correct. By using trigonometric identities and the given value of

\cos \theta

, the expression simplifies to

\left(\frac{{\sqrt{{3}}-1}}{{2}}+\frac{{1}}{{\sqrt{{2}}}}\right) \frac{{23}}{{17}}

23 Hard

23 of 26

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.

24 Hard

24 of 26

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).

25 Hard

25 of 26

Understanding +4 | -1

The solution set of inequality

\cos x \geq-\frac{1}{2}

View answer and explanation

Correct option: A

\bigcup\limits_{{n \in Z}}\left[2 n \pi-\frac{2 \pi}{3}, 2 n \pi+\frac{2 \pi}{3}\right]

Option A (

\bigcup\limits_ {{n \in Z}}\left[2 n \pi-\frac{{2 \pi}}{3}, 2 n \pi+\frac{{2 \pi}}{3}\right]

) is correct because

\cos x \geq -1/2

when

x

is between

-2\pi/3

and

2\pi/3

in each period.

26 Hard

26 of 26

Application +4 | -1

In a triangle

A B C

, the angle

B

is greater than the angle

A

. If the values of angles

A

and

B

satisfy the equation

3 \sin x-4 \sin ^{3} x-k=0,0 < k < 1

, then the values of

C

View answer and explanation

Correct option: C

2 \pi / 3

Option C is correct because it is derived from the sine triple angle identity and the given conditions, leading to the correct value for angle C.

Revision notes and key takeaways

Review TRIGONOMETRIC EQUATIONS in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 26 single-correct-answer MCQs on TRIGONOMETRIC EQUATIONS.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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