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

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

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

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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Medium

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Showing 241-270 of 294 questions

241 Medium

241 of 294

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

242 Medium

242 of 294

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})

243 Medium

243 of 294

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.

244 Medium

244 of 294

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.

245 Medium

245 of 294

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.

246 Medium

246 of 294

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.

247 Medium

247 of 294

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}

248 Medium

248 of 294

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).

249 Medium

249 of 294

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.

250 Medium

250 of 294

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}}

251 Medium

251 of 294

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.

252 Medium

252 of 294

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.

253 Medium

253 of 294

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

254 Medium

254 of 294

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}]

255 Medium

255 of 294

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.

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256 Medium

256 of 294

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.

257 Medium

257 of 294

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].

258 Medium

258 of 294

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.

259 Medium

259 of 294

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.

260 Medium

260 of 294

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}

261 Medium

261 of 294

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.

262 Medium

262 of 294

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.

263 Medium

263 of 294

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

264 Medium

264 of 294

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.

265 Medium

265 of 294

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.

266 Medium

266 of 294

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.

267 Medium

267 of 294

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.

268 Medium

268 of 294

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

269 Medium

269 of 294

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.

270 Medium

270 of 294

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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