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

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

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

The solution set of $(5 + 4\,\,\cos \,\,{\rm{\theta }})\,(2\,\,\cos \,\,{\rm{\theta }} + 1) = 0$ in the interval $\left[ {0,\;\,2{\rm{\pi }}} \right]$ is

$\left\{ {\frac{{2{\rm{\pi }}}}{3},\,\,\frac{{4{\rm{\pi }}}}{3}} \right\}$

$\left\{ {\frac{{\rm{\pi }}}{3},\,\,\frac{{2{\rm{\pi }}}}{3}} \right\}$

$\left\{ {\frac{{\rm{\pi }}}{3},{\rm{\;\pi }}} \right\}$

$\left\{ {\frac{{2{\rm{\pi }}}}{3},\,\,\frac{{5{\rm{\pi }}}}{3}} \right\}$

If $y = \frac{{\sin \,\,3\theta }}{{\sin \,\,\theta }},\;\,\theta \ne n\;\pi $, then

$y \in \left[ { - \,\infty ,\; - 1} \right]$

$y \in \left( {3,\;\,\infty } \right)$

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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Mixed difficulty

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511 Medium

511 of 581

Application +4 | -1

The general solution of

\tan \,\,\left( {\frac{\pi }{2}\,\,\sin \,\,\theta } \right) = \cot \left( {\frac{\pi }{2}\,\,\cos \,\,\theta } \right)

View answer and explanation

Correct option: B

\theta = 2r\;\pi ,\,\,r \in Z

Option B (

\theta = 2r\pi, r \in Z

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

512 Medium

512 of 581

Application +4 | -1

\triangle A B C, \tan \frac{A}{2}, \tan \frac{B}{2}

and

\tan \frac{C}{2}

are in HP, then the value of

\cot \frac{A}{2} \cot \frac{C}{2}

View answer and explanation

Correct option: C

Option C (3) is correct because when

\tan \frac{A}{2}, \tan \frac{B}{2}

and

\tan \frac{C}{2}

are in HP,

\cot \frac{A}{2} \cot \frac{C}{2}

equals 3.

513 Medium

513 of 581

Application +4 | -1

If in a triangle

\cos 3 \mathrm{~A}+\cos 3 \mathrm{~B}+\cos 3 \mathrm{C}=1

then one angle must be exactly equal to -

View answer and explanation

Correct option: D

120^{\circ}

Option D (

120^{\circ}

) is correct. By using trigonometric identities and the given equation, we can deduce that one of the angles must be

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

, which is

120^{\circ}

514 Easy

514 of 581

Application +4 | -1

x^{2}-4 x+5-\sin y=0, y \in[0,2 \pi]

, then

View answer and explanation

Correct option: D

x=2, y=\pi / 2

Option D is correct because it satisfies the given equation when the left-hand side is minimized and the right-hand side is maximized, both equaling 1.

515 Medium

515 of 581

Application +4 | -1

\sin \theta_1-\sin \theta_2=a

and

\cos \theta_1+\cos \theta_2=b

, then

View answer and explanation

Correct option: B

a^2+b^2 \leq 4

Option B (

a^2+b^2 \leq 4

) is correct because by squaring and adding the given equations, and using trigonometric identities, we find that

a^2 + b^2 = 2 + 2\cos(\theta_1 + \theta_2)

, which is always less than or equal to 4.

516 Easy

516 of 581

Application +4 | -1

The general solution of the equation

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

View answer and explanation

Correct option: A

x=2 n \pi+\frac{\pi}{4} \pm {A}

Option A (

x=2 n \pi+\frac{{\pi}}{{4}} \pm {A}

) is correct because it is derived by transforming the equation into a cosine addition formula.

517 Easy

517 of 581

Application +4 | -1

\sin 32^{\circ}=k

and

\cos x=1-2 k^2 ; \alpha, \beta

are the values of

x

between

0^{\circ}

and

360^{\circ}

with

\alpha < \beta

, then

View answer and explanation

Correct option: C

\beta=4 \alpha+40^{\circ}

Option C (

\beta=4 \alpha+40^{\circ}

) is correct because by solving for x and substituting the values of alpha and beta, the equation holds true.

518 Easy

518 of 581

Application +4 | -1

2\,\,\sec \,\,2\alpha = \tan \,\,\beta + \cot \,\,\beta

, then one of the values of

\alpha + \beta

View answer and explanation

Correct option: A

\frac{\pi }{4}

Option A (

\frac{\pi}{4}

) is correct because it is one of the possible values obtained after simplifying the given trigonometric equation and solving for

\alpha + \beta

519 Hard

519 of 581

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.

520 Easy

520 of 581

Analyzing +4 | -1

The solution of the equation

{\cos ^2}x - 2\cos x = 4\sin x - \sin \,\,2x\,\,\left( {0 \le x \le \pi } \right)

View answer and explanation

Correct option: C

\pi + {\tan ^{ - 1}}\left( { - \frac{1}{2}} \right)

Option D (

\pi + {\tan ^{ - 1}}\left( { - \frac{1}{2}} \right)

) is correct because by simplifying the given equation, we find that

\tan x = -1/2

and

x = \pi + {\tan ^{ - 1}}\left( { - \frac{1}{2}} \right)

in the given range.

521 Easy

521 of 581

Application +4 | -1

The equation

3 \cos x+4 \sin x=6

has

View answer and explanation

Correct option: D

No solution

Option D (No solution) is correct because the maximum value of

3 \cos x + 4 \sin x

is 5, and the equation

3 \cos x + 4 \sin x = 6

has no solution since 6 is outside the range [-5, 5].

522 Medium

522 of 581

Application +4 | -1

The most general solutions of the equation

\sec \,x - 1 = \left( {\sqrt 2 - 1} \right)\tan x

are given by

View answer and explanation

Correct option: B

2\,n\pi ,\;\,2\,n\pi + \frac{\pi }{4}

Option B is correct because it represents the general solutions obtained by solving the given trigonometric equation.

523 Easy

523 of 581

Understanding +4 | -1

The equation

\sin x + siny + \sin z = - 3

for

0 \le x \le 2\pi ,\;\,0 \le y \le 2\pi ,\;\,0 \le z\,\, \le \,\,2\pi

has

View answer and explanation

Correct option: A

One solution

Option A (One solution) is correct because the minimum value of sin(x) is -1, and the equation is only satisfied when sin(x) = sin(y) = sin(z) = -1.

524 Medium

524 of 581

Analyzing +4 | -1

The number of values of

\alpha

[0,2 \pi]

for which

2 \sin ^{3} \alpha-7 \sin ^{2} \alpha+7 \sin \alpha=2

, is:

View answer and explanation

Correct option: C

Option C (3) is correct because solving the equation

2 \sin ^{3} \alpha-7 \sin ^{2} \alpha+7 \sin \alpha=2

gives three solutions in the interval [0, 2

\pi

525 Easy

525 of 581

Understanding +4 | -1

Assertion: The inequality

3 x+2 y \geq 5

is the linear inequality.

Reason: The solution of

5 x-3 < 7

, when

x

is a real number, is

(-\infty, 2)

View answer and explanation

Correct option: B

Assertion is correct, reason is correct; reason is not a
correct explanation for assertion

Option C (Both are true) is correct because both the given identities involving combinations are standard and correct.

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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

526 Medium

526 of 581

Application +4 | -1

The general solution for

\alpha

is, if

\sin ^2 \alpha-\cos \alpha=1 / 4

for

0 \leq \alpha \leq 2 \pi

View answer and explanation

Correct option: D

\frac{\pi}{3}, \frac{5 \pi}{3}

Option D (

\frac{{\pi}}{{3}}, \frac{{5 \pi}}{{3}}

) is correct because these are the solutions to the trigonometric equation within the given interval.

527 Medium

527 of 581

Analyzing +4 | -1

The number of solutions of the equation

2\,\,\cos \,\,\left( {{e^x}} \right) = {5^x} + {5^{ - x}}

are

View answer and explanation

Correct option: A

No solution

Option A (No solution) is correct because the equation leads to a contradiction, where the LHS is always less than or equal to 2, while the RHS is always greater than or equal to 2, and they are not equal at x=0.

528 Easy

528 of 581

Application +4 | -1

The solution set of

(5 + 4\,\,\cos \,\,\mathrm{\theta })\,(2\,\,\cos \,\,\mathrm{\theta } + 1) = 0

in the interval

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

View answer and explanation

Correct option: C

\left\{ {\frac{{2\mathrm{\pi }}}{3},\,\,\frac{{4\mathrm{\pi }}}{3}} \right\}

Option C is correct because the solution set is found by solving the trigonometric equation and considering the interval [0, 2π].

529 Medium

529 of 581

Application +4 | -1

1+\sin x+\sin ^2 x+\sin ^3 x+\ldots \infty

is equal to

4+2 \sqrt{3}, 0 < x < \pi

, then

x

is equal to:

View answer and explanation

Correct option: D

\frac{\pi}{3}

\frac{2 \pi}{3}

Option D (

\frac{\pi}{3}

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

) is correct because these are the two possible values of x within the given range that satisfy the infinite geometric series equation.

530 Easy

530 of 581

Application +4 | -1

\operatorname{cosec} A+\cot A=\frac{11}{2}

, then

\tan A

is equal to

View answer and explanation

Correct option: C

\frac{44}{117}

Option C (

\frac{{44}}{{117}}

) is correct because by using the identity

\csc^2 A - \cot^2 A = 1

and solving the system of equations, we can find the value of

\tan A

531 Easy

531 of 581

Analyzing +4 | -1

The equation

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

possesses solution, if

View answer and explanation

Correct option: B

2 \le k \le 6

Option B (

2 \le k \le 6

) is correct because the solution is obtained by simplifying the given trigonometric equation and applying the range of the sine function.

532 Medium

532 of 581

Analyzing +4 | -1

The value of

\cos y \cos \left(\frac{\pi}{2}-x\right)-\cos \left(\frac{\pi}{2}-y\right) \cos x

+\sin y \cos \left(\frac{\pi}{2}-x\right)+\cos x \sin \left(\frac{\pi}{2}-y\right)

is zero, if

View answer and explanation

Correct option: D

x=n \pi-\frac{\pi}{4}+y

Option D (

x=n \pi-\frac{\pi}{4}+y

) is correct because it is derived from simplifying the trigonometric expression and solving for x in terms of y.

533 Easy

533 of 581

Application +4 | -1

The value of expression

\frac{1-4 \sin 10^{\circ} \sin 70^{\circ}}{2 \sin 10^{\circ}}

View answer and explanation

Correct option: B

Option B (1) is correct because using the product-to-sum trigonometric identity, the expression simplifies to 1.

534 Medium

534 of 581

Analyzing +4 | -1

y=\sin ^2 \theta+\operatorname{cosec}^2 \theta, \theta \neq 0

, then

View answer and explanation

Correct option: D

y \geq 2

Option D (

y \geq 2

) is correct because the minimum value of

\sin^2 \theta + \csc^2 \theta

is 2, which occurs when

\sin^2 \theta = 1

535 Medium

535 of 581

Application +4 | -1

If for two regular polygons, the ratio of the number of sides is

5: 4

and dfference of the measures of internal angles in radians is

\frac{\pi}{30}

, then the number of sides of both the polygons are

\ldots . .

View answer and explanation

Correct option: A

15,12

Option B (

\sec A - \tan A

) is correct because simplifying the given expression using trigonometric identities and the given range for A leads to this result.

536 Easy

536 of 581

Analyzing +4 | -1

y = \frac{{\sin \,\,3\theta }}{{\sin \,\,\theta }},\;\,\theta \ne n\;\pi

, then

View answer and explanation

Correct option: D

y \in \left[ { - 1,\,\;3} \right)

Option D (

y \in \left[ { - 1,\;3} \right)

) is correct because it represents the range of the function

y = \frac{{\sin \,3\theta }}{{\sin \,\theta }}

for

\theta \ne n\;\pi

537 Easy

537 of 581

Analyzing +4 | -1

Total number of solutions of

{\sin ^4}x + {\cos ^4}x = \sin x\,\, \cdot \,\,\cos x\,\;\mathrm{in}\,\;\left[ {0,\;\,2\pi } \right]

is equal to

View answer and explanation

Correct option: A

Option A (2) is correct because by simplifying the given equation, we find that there are only two solutions for x in the interval [0, 2

\pi

538 Easy

538 of 581

Application +4 | -1

1+\sin \theta+\sin ^2 \theta+\ldots \ldots \infty=4+2 \sqrt{3}, 0<\theta<\pi, \theta \neq \frac{\pi}{2}

, then

View answer and explanation

Correct option: D

\theta=\frac{\pi}{3}

\frac{2 \pi}{3}

Option D (

\theta = \frac{\pi}{3}

\frac{2\pi}{3}

) is correct because it is the solution obtained by using the formula for the sum of an infinite geometric series and solving for

\sin \theta

539 Easy

539 of 581

Application +4 | -1

The solution set of the inequality

3(2-x) \geq 2(1-x)

(-\infty, a]

. The value of

a

View answer and explanation

Correct option: A

Option A (4) is correct because solving the inequality

3(2-x) \geq 2(1-x)

results in

x \leq 4

, which corresponds to the interval

(-\infty, 4]

540 Medium

540 of 581

Analyzing +4 | -1

The general solution of the equation

\left(\cos \frac{x}{4}-2 \sin x\right) \sin x +\left(1+\sin \frac{x}{4}-2 \cos x\right) \cos x=0 \text { is }

View answer and explanation

Correct option: A

2 \pi+8 m \pi

Option A (

2 \pi+8 m \pi

) is correct because it represents the general solution where both

\sin \left(\frac{{5 x}}{{4}}\right)=1

and

\cos x=1

are simultaneously satisfied.

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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