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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Easy

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211 Easy
211 of 261
Application +4 | -1
The solution of sin⁡  x+sin⁡  5x=sin⁡  3x\sin \,\,x + \sin \,\,5x = \sin \,\,3x in (0,   π2)\left( {0,\;\,\frac{\pi }{2}} \right)

View answer and explanation
Correct option: B
π6,   π3\frac{\pi }{6},\;\,\frac{\pi }{3}
Option B (π6,   π3\frac{\pi }{6},\;\,\frac{\pi }{3}) is correct because these are the solutions to the trigonometric equation within the given interval after simplification.
212 Easy
212 of 261
Application +4 | -1
The smallest positive angle which satisfies the equation 2sin⁡2θ+3cos⁡θ+1=02 \sin ^2 \theta+\sqrt{3} \cos \theta+1=0 is
View answer and explanation
Correct option: A
5π6\frac{5 \pi}{6}
Option A (5π6\frac{{5 \pi}}{{6}}) is correct because it is the smallest positive angle that satisfies the given trigonometric equation after simplification and solving for θ\theta.
213 Easy
213 of 261
Analyzing +4 | -1
If tan⁡(πcos⁡θ)=cot⁡(πsin⁡θ)\tan (\pi \cos \theta)=\cot (\pi \sin \theta), then the value(s) of cos⁡(θ−π4)\cos \left(\theta-\frac{\pi}{4}\right) is/are
View answer and explanation
Correct option: C
±122\pm \frac{1}{2 \sqrt{2}}
Option C, ±122\pm \frac{{1}}{{2 \sqrt{{2}}}} is correct because it represents the possible values of cos⁡(θ−π4)\cos \left(\theta-\frac{{\pi}}{{4}}\right) obtained by solving the given trigonometric equation.
214 Easy
214 of 261
Application +4 | -1
If 1+cot⁡θ≤π21+\cot \theta \leq \frac{\pi}{2}, for 0<θ<π0 < \theta < \pi, then θ\theta is equal to
View answer and explanation
Correct option: C
π2\frac{\pi}{2}
Option C (π2\frac{{\pi}}{{2}}) is correct because it satisfies the inequality 1+cot⁡θ≤cot⁡θ21 + \cot \theta \le \cot \frac{{\theta}}{{2}} for 0<θ<π0 < \theta < \pi.
215 Easy
215 of 261
Application +4 | -1
If sin⁡αsin⁡β−cos⁡αcos⁡β+1=0\sin \alpha \sin \beta-\cos \alpha \cos \beta+1=0, then the value of cot⁡αtan⁡β\cot \alpha \tan \beta is
View answer and explanation
Correct option: A
−1-1
Option A (-1) is correct because the given condition implies that α+β=2nπ\alpha + \beta = 2n\pi, and thus cot⁡αtan⁡β=−1\cot \alpha \tan \beta = -1.
216 Easy
216 of 261
Application +4 | -1
x=tan⁡−13+tan⁡−12x=\tan ^{-1} 3+\tan ^{-1} 2,   y=tan⁡−13−tan⁡−12\,\,y=\tan ^{-1} 3-\tan ^{-1} 2 If atan⁡x+btan⁡y=0a \tan x+b \tan y=0, then a+ba+b is equal to kak a where kk is equal to
View answer and explanation
Correct option: D
8
The correct answer is D (8) because solving for tan⁡x\tan x and tan⁡y\tan y and substituting into the given equation allows us to find the value of k.
217 Easy
217 of 261
Application +4 | -1
If sin⁡  (π  cos⁡  θ)=cos⁡  (π  sin⁡  θ)\sin \,\,\left( {\pi \,\,\cos \,\,\theta } \right) = \cos \,\,\left( {\pi \,\,\sin \,\,\theta } \right), then the value of cos⁡  (θ+π4)\cos \,\,\left( {\theta + \frac{\pi }{4}} \right) equals
View answer and explanation
Correct option: B
12 2\frac{1}{{2\,\sqrt 2 }}
Option B (122\frac{1}{2\sqrt 2 }) is correct because by using trigonometric identities, the equation simplifies to cos⁡(θ+π4)=122\cos(\theta + \frac{\pi}{4}) = \frac{1}{2\sqrt 2 }.
218 Easy
218 of 261
Application +4 | -1
cot⁡  θ=sin⁡  2  θ,  θ≠nπ,  n∈Z\cot \,\,\theta = \sin \,\,2\;\theta ,\,\,\theta \ne n\pi ,\,\,n \in Z, if θ\theta equals
View answer and explanation
Correct option: A
45∘45^\circ or 90∘90^\circ
Option A (45∘45^\circ or 90∘90^\circ) is correct because these are the solutions to the equation cot⁡θ=sin⁡2θ\cot \theta = \sin 2\theta within the given constraints.
219 Easy
219 of 261
Analyzing +4 | -1
If cos⁡x=2cos⁡y−12−cos⁡y\cos x=\frac{2 \cos y-1}{2-\cos y}, where x,y∈(0,π)x, y \in(0, \pi), then tan⁡x2cot⁡y2\tan \frac{x}{2} \cot \frac{y}{2} is equal to
View answer and explanation
Correct option: B
3\sqrt{3}
Option B (3\sqrt{3}) is correct because by substituting the given expression for cos⁡x\cos x into the half-angle formula, we can simplify and find the value of tan⁡x2cot⁡y2\tan \frac{x}{2} \cot \frac{y}{2}.
220 Easy
220 of 261
Application +4 | -1
If tan⁡(α π4)=cot⁡(β π4)\tan \left( {\frac{{\alpha \,\pi }}{4}} \right) = \cot \left( {\frac{{\beta \,\pi }}{4}} \right), then
View answer and explanation
Correct option: D
α+β=2(2n+1),  ∀n\alpha + \beta = 2\left( {2n + 1} \right),\;\forall n is an integer
Option D (α+β=2(2n+1),;∀n\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 α+β\alpha + \beta must be an even multiple of an odd number.
221 Easy
221 of 261
Application +4 | -1
The general value of θ\theta in the equation cos⁡  θ=12,  tan⁡  θ=−1\cos \,\,\theta = \frac{1}{{\sqrt 2 }},\,\,\tan \,\,\theta = - 1 is
View answer and explanation
Correct option: B
2nπ  ±  7π4,   n∈I2n\pi \,\, \pm \,\,\frac{{7\pi }}{4},\;\,n \in I
Option B (2nπ±7π4,  n∈I2n\pi \pm \frac{{7\pi }}{{4}},\;n \in I) is correct because it represents the general solution for θ\theta that satisfies both cos⁡θ=12\cos \theta = \frac{1}{\sqrt{2}} and tan⁡θ=−1\tan \theta = -1.
222 Easy
222 of 261
Application +4 | -1
The general solution of sin⁡  x−cos⁡  x=2\sin \,\,x - \cos \,\,x = \sqrt 2, for any integer nn is
View answer and explanation
Correct option: B
2nπ+3π42n\pi + \frac{{3\pi }}{4}
Option B (2nπ+3π42n\pi + \frac{{3\pi }}{{4}}) is correct because it represents the general solution to the trigonometric equation sin⁡x−cos⁡x=2\sin x - \cos x = \sqrt{2} after simplification and solving for x.
223 Easy
223 of 261
Analyzing +4 | -1
If 4  cos⁡  θ−3  sec⁡  θ=2tan⁡  θ,  then  θ4\,\,\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
nπ+(−1)n  π10n\pi + {\left( { - 1} \right)^n}\,\,\frac{\pi }{{10}}
Option A (nπ+(−1)nπ10n\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 θ\theta.
224 Easy
224 of 261
Analyzing +4 | -1
If u=a2cos⁡2θ+b2sin⁡2θ+a2sin⁡2θ+b2cos⁡2θu=\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 u2u^{2} is given by
View answer and explanation
Correct option: A
(a−b)2(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.
225 Easy
225 of 261
Analyzing +4 | -1
If 1+sin⁡2x1−sin⁡2x=cot⁡2(a+x)\frac{1+\sin 2 x}{1-\sin 2 x}=\cot ^2(a+x), ∀ x∈R∼(nπ+π4),n∈N\forall \, x \in R \sim\left(n \pi+\frac{\pi}{4}\right), n \in N, then aa is equal to
View answer and explanation
Correct option: C
3π4\frac{3 \pi}{4}
Option C (3π4\frac{3 \pi}{4}) is correct because by simplifying the given expression and using trigonometric identities, we can find that a=3π4a = \frac{3 \pi}{4}.

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226 Easy
226 of 261
Application +4 | -1
If tan⁡9θ=34\tan 9 \theta=\frac{3}{4} where (0<θ<π18)\left(0<\theta<\frac{\pi}{18}\right) Then value of 3cosec⁡3θ−4sec⁡3θ=3 \operatorname{cosec} 3 \theta-4 \sec 3 \theta=
View answer and explanation
Correct option: D
10
Option D (10) is correct because by using trigonometric identities and the given information, the expression simplifies to 10.
227 Easy
227 of 261
Application +4 | -1
If 0≤x≤π0 \leq \mathrm{x} \leq \pi, then the solution of the equation 16sin⁡2x+16cos⁡2x−1016^{\sin ^2 x}+16^{\cos ^2 x}-10 is given by xx equal to-
View answer and explanation
Correct option: A
π/6,π/3\pi / 6, \pi / 3
Option A (π/6,π/3\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.
228 Easy
228 of 261
Analyzing +4 | -1
The set of values of xx for which sin⁡xcos⁡3x>cos⁡xsin⁡3x,0≤x≤2π\sin x \cos ^3 x>\cos x \sin ^3 x, 0 \leq x \leq 2 \pi, is
View answer and explanation
Correct option: B
(0,π4)\left(0, \frac{\pi}{4}\right)
Option B ((0,π4)\left(0, \frac{\pi}{4}\right)) is correct because it represents the interval where the inequality sin⁡xcos⁡3x>cos⁡xsin⁡3x\sin x \cos ^3 x > \cos x \sin ^3 x holds true for 0≤x≤2π0 \leq x \leq 2 \pi.
229 Easy
229 of 261
Application +4 | -1
Let SS be the set of all α∈R\alpha \in R such that the equation, cos⁡2x+αsin⁡x=2α−7\cos 2x + \alpha \sin x = 2\alpha - 7 has a solution. Then, SS 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 α\alpha for which the given equation has a solution, derived by analyzing the range of the sine function.
230 Easy
230 of 261
Application +4 | -1
If x2−4x+5−sin⁡y=0,y∈[0,2π]x^{2}-4 x+5-\sin y=0, y \in[0,2 \pi], then
View answer and explanation
Correct option: D
x=2,y=π/2x=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.
231 Easy
231 of 261
Application +4 | -1
The general solution of the equation sin⁡x+cos⁡x=2cos⁡A\sin x+\cos x=\sqrt{2} \cos {A} is
View answer and explanation
Correct option: A
x=2nπ+π4±Ax=2 n \pi+\frac{\pi}{4} \pm {A}
Option A (x=2nπ+π4±Ax=2 n \pi+\frac{{\pi}}{{4}} \pm {A}) is correct because it is derived by transforming the equation into a cosine addition formula.
232 Easy
232 of 261
Application +4 | -1
If sin⁡32∘=k\sin 32^{\circ}=k and cos⁡x=1−2k2;α,β\cos x=1-2 k^2 ; \alpha, \beta are the values of xx between 0∘0^{\circ} and 360∘360^{\circ} with α<β\alpha < \beta, then
View answer and explanation
Correct option: C
β=4α+40∘\beta=4 \alpha+40^{\circ}
Option C (β=4α+40∘\beta=4 \alpha+40^{\circ}) is correct because by solving for x and substituting the values of alpha and beta, the equation holds true.
233 Easy
233 of 261
Application +4 | -1
If 2  sec⁡  2α=tan⁡  β+cot⁡  β2\,\,\sec \,\,2\alpha = \tan \,\,\beta + \cot \,\,\beta, then one of the values of α+β\alpha + \beta is
View answer and explanation
Correct option: A
π4\frac{\pi }{4}
Option A (π4\frac{\pi}{4}) is correct because it is one of the possible values obtained after simplifying the given trigonometric equation and solving for α+β\alpha + \beta.
234 Easy
234 of 261
Analyzing +4 | -1
The solution of the equation cos⁡2x−2cos⁡x=4sin⁡x−sin⁡  2x  (0≤x≤π){\cos ^2}x - 2\cos x = 4\sin x - \sin \,\,2x\,\,\left( {0 \le x \le \pi } \right) is
View answer and explanation
Correct option: C
π+tan⁡−1(−12)\pi + {\tan ^{ - 1}}\left( { - \frac{1}{2}} \right)
Option D (π+tan⁡−1(−12)\pi + {\tan ^{ - 1}}\left( { - \frac{1}{2}} \right)) is correct because by simplifying the given equation, we find that tan⁡x=−1/2\tan x = -1/2 and x=π+tan⁡−1(−12)x = \pi + {\tan ^{ - 1}}\left( { - \frac{1}{2}} \right) in the given range.
235 Easy
235 of 261
Application +4 | -1
The equation 3cos⁡x+4sin⁡x=63 \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 3cos⁡x+4sin⁡x3 \cos x + 4 \sin x is 5, and the equation 3cos⁡x+4sin⁡x=63 \cos x + 4 \sin x = 6 has no solution since 6 is outside the range [-5, 5].
236 Easy
236 of 261
Understanding +4 | -1
The equation sin⁡x+siny+sin⁡z=−3\sin x + siny + \sin z = - 3 for 0≤x≤2π,   0≤y≤2π,   0≤z  ≤  2π0 \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.
237 Easy
237 of 261
Understanding +4 | -1
Assertion: The inequality 3x+2y≥53 x+2 y \geq 5 is the linear inequality.

Reason: The solution of 5x−3<75 x-3 < 7, when xx is a real number, is (−∞,2)(-\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.
238 Easy
238 of 261
Application +4 | -1
The solution set of (5+4  cos⁡  θ) (2  cos⁡  θ+1)=0(5 + 4\,\,\cos \,\,\mathrm{\theta })\,(2\,\,\cos \,\,\mathrm{\theta } + 1) = 0 in the interval [0,   2π]\left[ {0,\;\,2\mathrm{\pi }} \right] is

View answer and explanation
Correct option: C
{2π3,  4π3}\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].
239 Easy
239 of 261
Application +4 | -1
If cosec⁡A+cot⁡A=112\operatorname{cosec} A+\cot A=\frac{11}{2}, then tan⁡A\tan A is equal to
View answer and explanation
Correct option: C
44117\frac{44}{117}
Option C (44117\frac{{44}}{{117}}) is correct because by using the identity csc⁡2A−cot⁡2A=1\csc^2 A - \cot^2 A = 1 and solving the system of equations, we can find the value of tan⁡A\tan A.
240 Easy
240 of 261
Analyzing +4 | -1
The equation k  sin⁡  x+cos⁡  2  x=2  k−7k\,\,\sin \,\,x + \cos \,\,2\;x = 2\;k - 7 possesses solution, if
View answer and explanation
Correct option: B
2≤k≤62 \le k \le 6
Option B (2≤k≤62 \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.
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