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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Easy

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121 Easy
121 of 261
Application +4 | -1
If tan⁡2θ  tan⁡θ=1\tan 2\mathrm{\theta }\,\,\tan \mathrm{\theta } = 1, then the general value of θ\mathrm{\theta } is
View answer and explanation
Correct option: A
(  n+12)π3\left( {\;n + \frac{1}{2}} \right)\frac{\pi }{3}
Option A, (  n+12)π3\left( {\;n + \frac{{1}}{{2}}} \right)\frac{{\pi }}{{3}}, is the correct general solution for θ\theta derived from the given trigonometric equation by using trigonometric identities and solving for θ\theta.
122 Easy
122 of 261
Analyzing +4 | -1
The sum of all values of xx in [0,  2π]\left[ {0,\,\,2\pi } \right], for which sin⁡x+sin⁡2x+sin⁡3x+sin⁡4x=0\sin x + \sin 2x + \sin 3x + \sin 4x = 0, is equal to
View answer and explanation
Correct option: D
9 π9\,\pi
Option D (9π9\pi) is correct because the sum of all solutions to the trigonometric equation within the given interval [0, 2π\pi] is 9π\pi.
123 Easy
123 of 261
Application +4 | -1
The solution set of the equation 5cos⁡2θ+2cos⁡2θ2+15 \cos 2 \theta+2 \cos ^2 \frac{\theta}{2}+1 =0=0, in the interval −π<θ<π-\pi<\theta<\pi, is
View answer and explanation
Correct option: B
{π3,π−cos⁡−1(35)}\left\{\frac{\pi}{3}, \pi-\cos ^{-1}\left(\frac{3}{5}\right)\right\}
Option B is the correct solution set because it satisfies the given trigonometric equation within the specified interval.
124 Easy
124 of 261
Analyzing +4 | -1
If 0≤x<π20 \le x < \frac{\pi }{2}, then the number of values of xx for which x−sin⁡2x+sin⁡3x=0x - \sin 2x + \sin 3x = 0, is
View answer and explanation
Correct option: A
2
Option A (2) is correct because there are two values of x (0 and π3\frac{\pi}{3}) in the given interval that satisfy the equation.
125 Easy
125 of 261
Application +4 | -1
The set of real xx satisfying the inequality 5−2x3≤x6−5\frac{5-2 x}{3} \leq \frac{x}{6}-5 is [a,∞)[a, \infty). The value of aa is
View answer and explanation
Correct option: A
8
Option A (8) is correct because solving the inequality 5−2x3≤x6−5\frac{5-2x}{3} \leq \frac{x}{6} - 5 results in x≥8x \geq 8, which corresponds to the interval [8,∞)[8, \infty).
126 Easy
126 of 261
Application +4 | -1
The equation 3sin⁡2x+10cos⁡x−6=03 \sin ^2 x+10 \cos x-6=0 is satisfied if
View answer and explanation
Correct option: B
x=2nπ±cos⁡−1(13)x=2 n \pi \pm \cos ^{-1}\left(\frac{1}{3}\right)
Option B (x=2nπ±cos⁡−1(13)x=2 n \pi \pm \cos ^{-1}(\frac{1}{3})) is correct because it represents the general solution to the given trigonometric equation.
127 Easy
127 of 261
Analyzing +4 | -1
The general solution of :  sin⁡2θ  sec⁡  θ+3  tan⁡  θ=0:\,\,{\sin ^2}\theta \,\,\sec \,\,\theta + \sqrt 3 \,\,\tan \,\,\theta = 0 is
View answer and explanation
Correct option: B
θ=n  π,  n∈Z\theta = n\;\pi ,\,\,n \in Z
Option B (θ=n;π,;;n∈Z\theta = n\\;\pi ,\\;\\;n \in Z) is correct because the given equation simplifies to (sin⁡θ+3)tan⁡θ=0(\sin \theta + \sqrt 3 )\tan \theta = 0, leading to tan⁡θ=0\tan \theta = 0 and thus θ=n;π\theta = n\\;\pi.
128 Easy
128 of 261
Application +4 | -1
If nn is any integer, then the general solution of the equation cos⁡  θ−sin⁡  θ=12\cos \,\,\theta - \sin \,\,\theta = \frac{1}{{\sqrt 2 }} is
View answer and explanation
Correct option: C
θ=2 nπ+π12\theta = 2\,n\pi + \frac{\pi }{{12}} or θ=2 nπ−7π12\theta = 2\,n\pi - \frac{{7\pi }}{{12}}
Option D is correct because by rewriting the equation and using trigonometric identities, we can find the general solution for θ\theta.
129 Easy
129 of 261
Application +4 | -1
The number of values of xx in [0,  5  π]\left[ {0,\,\,5\;\pi } \right] satisfying the equation 3  cos⁡  2x−10  cos⁡  x+7=03\,\,\cos \,\,2x - 10\,\,\cos \,\,x + 7 = 0 is
View answer and explanation
Correct option: C
8
Option A (5) is incorrect. The correct answer is 8 because there are 3 solutions for cos(x) = 1 and 5 solutions for cos(x) = 2/3 in the given interval.
130 Easy
130 of 261
Application +4 | -1
In △ABC\triangle A B C, if ∠C=2π3\angle C=\frac{2 \pi}{3}, then the value of cos⁡2A+cos⁡2B−cos⁡Acos⁡B\cos ^2 A+\cos ^2 B-\cos A \cos B is
View answer and explanation
Correct option: A
34\frac{3}{4}
Option A (34\frac{3}{4}) is correct because by using the given angle and trigonometric identities, the expression simplifies to 34\frac{3}{4}.
131 Easy
131 of 261
Application +4 | -1
If cot⁡α\cot \alpha equals the integral solution of the inequality 4x2−16x+15<04 x^2-16 x+15<0 and sin⁡β\sin \beta equals to the slope of the bisector of the first quadrant, then sin⁡(α+β)sin⁡(α−β)\sin (\alpha+\beta) \sin (\alpha-\beta) is equal to
View answer and explanation
Correct option: B
−4/5-4 / 5
Option B (-4/5) is correct because by solving the inequality and using the given information, we can find the values of cot⁡α\cot \alpha and sin⁡β\sin \beta, and then calculate sin⁡(α+β)sin⁡(α−β)\sin(\alpha+\beta)\sin(\alpha-\beta) using trigonometric identities.
132 Easy
132 of 261
Application +4 | -1
If cos⁡−1(x/2)+cos⁡−1(y/3)=θ\cos ^{-1}(x / 2)+\cos ^{-1}(y / 3)=\theta, then 9x2−12xycos⁡θ9 x^2-12 x y \cos \theta +4y2+4 y^2 is equal to
View answer and explanation
Correct option: C
36sin⁡2θ36 \sin ^2 \theta
Option C, 36sin⁡2θ36 \sin^2 \theta, is correct because by using the cosine addition formula and simplifying, we can derive that 9x2−12xycos⁡θ+4y2=36sin⁡2θ9x^2 - 12xy \cos \theta + 4y^2 = 36 \sin^2 \theta.
133 Easy
133 of 261
Application +4 | -1
If tan⁡  (πcos⁡  θ)=cot⁡(π sin⁡θ),   0<θ<3π4\tan \,\,\left( {\pi \cos \,\,\theta } \right) = \cot \left( {\pi \,\sin \theta } \right),\;\,0 < \theta < \frac{{3\pi }}{4}, then sin⁡(θ+π4)\sin \left( {\theta + \frac{\pi }{4}} \right) equals

View answer and explanation
Correct option: C
122\frac{1}{{2\sqrt 2 }}
Option C (122\frac{1}{{2\sqrt 2 }}) is correct because solving the given equation leads to sin⁡(θ+π4)=122\sin(\theta + \frac{\pi}{4}) = \frac{1}{{2\sqrt 2 }}.
134 Easy
134 of 261
Application +4 | -1
If cot⁡xcot⁡y=k\cot x \cot y=k and x+y=π3x+y=\frac{\pi}{3}, then tan⁡x,tan⁡y\tan x, \tan y satisfy the equation
View answer and explanation
Correct option: A
kt2−3(k−1)t+1=0k t^2-\sqrt{3}(k-1) t+1=0
Option A (kt2−3(k−1)t+1=0k t^2-\sqrt{3}(k-1) t+1=0) is correct because it represents the quadratic equation whose roots are tan⁡x\tan x and tan⁡y\tan y, derived from the given conditions cot⁡xcot⁡y=k\cot x \cot y = k and x+y=π3x+y = \frac{\pi}{3}.
135 Easy
135 of 261
Application +4 | -1
Solution of the equation 3tan⁡  (θ−15)=tan⁡  (θ+15)3\tan \,\,\left( {\theta - 15} \right) = \tan \,\,\left( {\theta + 15} \right) is
View answer and explanation
Correct option: D
θ=nπ+π4\theta = n\pi + \frac{\pi }{4}
Option D (θ=nπ+π4\theta = n\pi + \frac{\pi }{4}) is correct because it represents the general solution to the given trigonometric equation, found by applying trigonometric identities and simplifying.

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136 Easy
136 of 261
Analyzing +4 | -1
If cos⁡  2x=(2+1)  (cos⁡  x−12),  cos⁡  x≠12\cos \,\,2x = \left( {\sqrt 2 + 1} \right)\,\,\left( {\cos \,\,x - \frac{1}{{\sqrt 2 }}} \right),\,\,\cos \,\,x \ne \frac{1}{2}, then x∈Ix \in I
View answer and explanation
Correct option: D
{2nπ  ±  π4:n∈Z}\left\{ {2n\pi \,\, \pm \,\,\frac{\pi }{4}:n \in Z} \right\}
Option D ({2nπ±π4:n∈Z}\{ {2n\pi \pm \frac{\pi }{4}:n \in Z} \}) is correct because it represents the general solution obtained after solving the given trigonometric equation.
137 Easy
137 of 261
Application +4 | -1
The number of the solutions of the equation sin⁡5xcos⁡3x\sin 5 x \cos 3 x =sin⁡6xcos⁡2x=\sin 6 x \cos 2 x in the interval [0,π][0, \pi], is
View answer and explanation
Correct option: C
5
The correct answer is 5 because there are 5 solutions to the given equation within the specified interval [0, π\pi], found by simplifying the equation and solving for x.
138 Easy
138 of 261
Analyzing +4 | -1
If f(θ)=∣sin⁡θ∣+∣cos⁡θ∣,θ∈Rf(\theta)=|\sin \theta|+|\cos \theta|, \theta \in R, then
View answer and explanation
Correct option: D
f(θ)∈[1,2]f(\theta) \in[1, \sqrt{2}]
Option D (f(θ)∈[1,2]f(\theta) \in[1, \sqrt{2}]) is correct because the function's range is determined by the minimum and maximum values of ∣sin⁡θ∣+∣cos⁡θ∣|\sin \theta| + |\cos \theta|.
139 Easy
139 of 261
Application +4 | -1
The general value of θ\theta,obtained from the equation cos⁡2θ=sin⁡α\cos 2 \theta=\sin \alpha, is
View answer and explanation
Correct option: A
θ=nπ±(π4−α2)\theta=n \pi \pm\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)
Option A (θ=nπ±(π4−α2)\theta=n \pi \pm\left(\frac{{\pi}}{{4}}-\frac{{\alpha}}{{2}}\right)) is correct because it is derived by converting sine to cosine and applying the general solution formula.
140 Easy
140 of 261
Application +4 | -1
If 1+sin⁡  θ+sin⁡2  θ+…1 + \sin \,\,\theta + {\sin ^2}\,\,\theta + \ldots to ∞=4+2 3,  0<θ<π,  θ≠π2\infty = 4 + 2\,\sqrt 3 ,\,\,0 < \theta < \pi ,\,\,\theta \ne \frac{\pi }{2}, then θ=\theta =
View answer and explanation
Correct option: D
π3    or, 2  π3\frac{\pi }{3}\;\,\,\mathrm{or},\,\frac{{2\;\pi }}{3}
Option D (π3\frac{\pi}{3} or 2π3\frac{2\pi}{3}) is correct because solving the infinite geometric series equation results in sin⁡θ=32\sin \theta = \frac{\sqrt{3}}{2}, which gives the solutions θ=π3\theta = \frac{\pi}{3} and θ=2π3\theta = \frac{2\pi}{3}.
141 Easy
141 of 261
Application +4 | -1
If x=sin⁡θ∣sin⁡θ∣,y=cos⁡θ∣cos⁡θ∣x=\sin \theta|\sin \theta|, y=\cos \theta|\cos \theta|, where 99π2≤θ≤50π\frac{99 \pi}{2} \leq \theta \leq 50 \pi, then
View answer and explanation
Correct option: D
y−x=1y-x=1
Option D (y−x=1y-x=1) is correct because given the range of θ\theta, x=−sin⁡2θx = -\sin^2 \theta and y=cos⁡2θy = \cos^2 \theta, thus y−x=cos⁡2θ+sin⁡2θ=1y - x = \cos^2 \theta + \sin^2 \theta = 1.
142 Easy
142 of 261
Application +4 | -1
General solution of the equation cot⁡  θ  −  tan⁡  θ=2\cot \,\,\mathrm{\theta }\,\, - \,\,\tan \,\,\mathrm{\theta } = 2 is
View answer and explanation
Correct option: B
nπ2+π8\frac{{n\pi }}{2} + \frac{\pi }{8}
Option B (nπ2+π8\frac{{n\pi }}{2} + \frac{\pi }{8}) is correct because by simplifying the given equation using trigonometric identities, we get tan⁡2θ=1\tan 2\theta = 1, which leads to the general solution θ=nπ2+π8\theta = \frac{{n\pi }}{2} + \frac{\pi }{8}.
143 Easy
143 of 261
Analyzing +4 | -1
The number of real solutions of sin⁡ex⋅cos⁡ex=2x−2+2−x−2\sin e^x \cdot \cos e^x=2^{x-2}+2^{-x-2} is
View answer and explanation
Correct option: A
0
Option A (0) is correct because the equation sin⁡ex⋅cos⁡ex=2x−2+2−x−2\sin e^x \cdot \cos e^x=2^{x-2}+2^{-x-2} has no real solutions since the LHS is less than or equal to 1/2 and the RHS is greater than or equal to 1/2, and equality cannot hold.
144 Easy
144 of 261
Application +4 | -1
The number of solutions of the equation 32sec⁡2x+1=10.3tan⁡2x3^{2 \sec ^2 x}+1=10.3^{\tan ^2 x} in the interval [0,2π][0,2 \pi] is
View answer and explanation
Correct option: C
4
Option C (4) is correct because solving the equation 32sec⁡2x+1=10⋅3tan⁡2x3^{2 \sec^2 x} + 1 = 10 \cdot 3^{\tan^2 x} leads to four solutions for xx in the interval [0,2π][0, 2\pi].
145 Easy
145 of 261
Application +4 | -1
The value of 2sin⁡23π4+2cos⁡2π4+2sec⁡2π32 \sin ^2 \frac{3 \pi}{4}+2 \cos ^2 \frac{\pi}{4}+2 \sec ^2 \frac{\pi}{3} is
View answer and explanation
Correct option: A
10
Option A (10) is correct because substituting the values of the trigonometric functions and simplifying the expression results in 10.
146 Easy
146 of 261
Application +4 | -1
The value of cot⁡x−tan⁡xcot⁡2x\frac{\cot x-\tan x}{\cot 2 x} is
View answer and explanation
Correct option: B
2
Option B (2) is correct because the expression simplifies to 2 using trigonometric identities for cotangent and tangent functions.
147 Easy
147 of 261
Application +4 | -1
Let S={θ∈[−2π,  2π]:2cos⁡2θ+3sin⁡θ=0}S = \left\{ {\theta \in \left[ { - 2\pi ,\,\,2\pi } \right]:2{{\cos }^2}\theta + 3\sin \theta = 0} \right\}, then the sum of the elements of SS is
View answer and explanation
Correct option: A
2π2\pi
Option A (2π2\pi) is correct because it is the sum of all solutions of the trigonometric equation within the given interval.
148 Easy
148 of 261
Application +4 | -1
Let nn be a fixed positive integer such that sin⁡π2n+cos⁡π2n=n2\sin \frac{\pi}{2 n}+\cos \frac{\pi}{2 n}=\frac{\sqrt{n}}{2}, then
View answer and explanation
Correct option: C
n=6n=6
Option C (n=6n=6) is correct because substituting n=6n=6 into the equation sin⁡π2n+cos⁡π2n=n2\sin \frac{\pi}{2 n}+\cos \frac{\pi}{2 n}=\frac{\sqrt{n}}{2} satisfies the equation, while other options do not.
149 Easy
149 of 261
Application +4 | -1
If sin⁡  (π4  cot⁡  θ)=cos⁡  (π4  tan⁡  θ)\sin \,\,\left( {\frac{\pi }{4}\,\,\cot \,\,\theta } \right) = \cos \,\,\left( {\frac{\pi }{4}\,\,\tan \,\,\theta } \right), then θ\theta is equal to
View answer and explanation
Correct option: D
nπ  +  π4n\pi \,\, + \,\,\frac{\pi }{4}
Option D (516\frac{5}{16}) is correct because by using trigonometric identities and known values of sine functions, the expression simplifies to 516\frac{5}{16}.
150 Easy
150 of 261
Application +4 | -1
If 1+sin⁡  x+sin⁡2  x+sin⁡3  x+…+…∞1 + \sin \,\,x + {\sin ^2}\,\,x + {\sin ^3}\,\,x + \ldots + \ldots \infty is equal to 4+23,   0<x<π4 + 2\sqrt 3 ,\,\;0 < x < \pi, then x=x =

View answer and explanation
Correct option: D
π3\frac{\pi }{3} or 2 π3\frac{{2\,\pi }}{3}
Option D (π3\frac{\pi }{3} or 2,π3\frac{{2\\,\pi }}{3}) is correct because these are the solutions to the trigonometric equation derived from the infinite geometric series.
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