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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Easy

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Showing 91-120 of 261 questions

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91 Easy
91 of 261
Application +4 | -1
If tan⁡  (απ4)=cot⁡  (βπ4)\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∈Z\alpha + \beta = 2\left( {2n + 1} \right),\;n \in Z
Option D (α+β=2(2n+1),  n∈Z\alpha + \beta = 2\left( {2n + 1} \right),\;n \in Z) is correct because by converting cotangent to tangent and using the general solution for tangent equations, we find the relationship between α\alpha and β\beta.
92 Easy
92 of 261
Application +4 | -1
The most general value of θ\theta which satisfy both the equations cos⁡  θ=−12\cos \,\,\theta = - \frac{1}{{\sqrt 2 }} and tan⁡  θ=1\tan \,\,\theta = 1, is
View answer and explanation
Correct option: A
2nπ+5π4,    n∈I2n\pi + \frac{{5\pi }}{4},\;\,\,n \in I
Option A is correct because it identifies the general solution that satisfies both the cosine and tangent conditions, considering the quadrant in which both conditions are met.
93 Easy
93 of 261
Analyzing +4 | -1
General value of θ\mathrm{\theta } satisfying the equation tan⁡2  θ+sec⁡  2θ=1{\tan ^2}\,\,\mathrm{\theta } + \sec \,\,2\mathrm{\theta } = 1 is
View answer and explanation
Correct option: B
mπ,   nπ    ±  π3m\pi ,\;\,n\pi \;\,\, \pm \,\,\frac{\pi }{3}
Option B (mπ,   nπ    ±  π3m\pi ,\;\,n\pi \;\,\,\pm \,\,\frac{\pi }{3}) is correct because it represents the general solutions obtained by solving the given trigonometric equation.
94 Easy
94 of 261
Application +4 | -1
If sin⁡  (π  cot⁡  θ)=cos⁡  (π  tan⁡  θ)\sin \,\,\left( {\pi \,\,\cot \,\,\theta } \right) = \cos \,\,\left( {\pi \,\,\tan \,\,\theta } \right) then cot⁡  2  θ\cot \,\,2\;\theta is equal to where n∈Zn \in Z
View answer and explanation
Correct option: B
n+14n + \frac{1}{4}
Option B (n+14n + \frac{1}{4}) is correct because it is the simplified form of the expression after applying trigonometric identities and solving for cot⁡2θ\cot 2\theta.
95 Easy
95 of 261
Application +4 | -1
f(θ)=sin⁡2θ+sin⁡2(θ+2π3)+sin⁡2(θ+4π3)f(\theta)=\sin ^2 \theta+\sin ^2\left(\theta+\frac{2 \pi}{3}\right)+\sin ^2\left(\theta+\frac{4 \pi}{3}\right), then f(π15)f\left(\frac{\pi}{15}\right) is equal to
View answer and explanation
Correct option: B
32\frac{3}{2}
Option B (32\frac{3}{2}) is correct because simplifying the given trigonometric expression results in a constant value of 32\frac{3}{2}, independent of θ\theta.
96 Easy
96 of 261
Analyzing +4 | -1
The number of solutions of the equation 2cos⁡x=∣ sin⁡  x ∣{2^{\cos x}} = \left| {\,\sin \,\,x\,} \right| in [−2  π,  2  π]\left[ { - 2\;\pi ,\,\,2\;\pi } \right], is
View answer and explanation
Correct option: D
4\mathrm{4}
Option D (4) is correct because the graphs of the two functions, y=2cos⁡xy = {2^{\cos x}} and y = \left| {\,\sin \,x\} \right|y = \left| {\,\sin \,x\} \right|, intersect at four points within the given interval [−2π,2π][-2\pi, 2\pi].
97 Easy
97 of 261
Understanding +4 | -1
If sin⁡  A=sin⁡  B,  cos⁡  A=cos⁡  B\sin \,\,A = \sin \,\,B,\,\,\cos \,\,A = \cos \,\,B, then the value of AA in terms of BB is
View answer and explanation
Correct option: C
2  n  π+B2\;n\;\pi + B
Option C (2  n  π+B2\;n\;\pi + B) is correct because it satisfies both the sine and cosine relations for all integers n.
98 Easy
98 of 261
Application +4 | -1
The number of solutions of 5cos⁡2θ−3sin⁡2θ+6sin⁡θcos⁡θ=75 \cos ^2 \theta-3 \sin ^2 \theta+6 \sin \theta \cos \theta=7 in the interval [0,2π][0,2 \pi] is
View answer and explanation
Correct option: C
0
Option C (0) is correct because by rewriting the equation using trigonometric identities, we find that the equation has no solutions in the given interval.
99 Easy
99 of 261
Application +4 | -1
The general solution of sin⁡  x−3  sin⁡  2x+sin⁡  3x=cos⁡  x −3  cos⁡  2x+cos⁡  3x\sin \,\,x - 3\,\,\sin \,\,2x + \sin \,\,3x = \cos \,\,x\, - 3\,\,\cos \,\,2x + \cos \,\,3x is
View answer and explanation
Correct option: B
nπ2+π8  \frac{{n\pi }}{2} + \frac{\pi }{8}\;
Option B is correct because by rearranging the equation and factoring, we can find the general solution for x.
100 Easy
100 of 261
Application +4 | -1
The value of tan⁡40∘+tan⁡20∘+3tan⁡20∘tan⁡40∘\tan 40^{\circ}+\tan 20^{\circ}+\sqrt{3} \tan 20^{\circ} \tan 40^{\circ} is
View answer and explanation
Correct option: D
3\sqrt{3}
Option D (3\sqrt{3}) is correct because the expression simplifies to 3\sqrt{3} using the tangent addition formula.
101 Easy
101 of 261
Application +4 | -1
The set of all xx in (−π,π)(-\pi, \pi) satisfying ∣4sin⁡x−1∣<5|4 \sin x-1| < \sqrt{5} is given by
View answer and explanation
Correct option: B
(π10,3π10)\left(\frac{\pi}{10}, \frac{3 \pi}{10}\right)
Option B (π10,3π10)\left(\frac{\pi}{10}, \frac{3 \pi}{10}\right) is correct because solving the inequality ∣4sin⁡x−1∣<5|4 \sin x - 1| < \sqrt{5} for xx in the interval (−π,π)(-\pi, \pi) yields this solution set.
102 Easy
102 of 261
Application +4 | -1
If sin⁡  θ=3  cos⁡  θ,  π<θ<0\sin \,\,\mathrm{\theta } = \sqrt 3 \,\,\cos \,\,\mathrm{\theta },\mathrm{\;\pi } < \mathrm{\theta } < 0, then θ\mathrm{\theta } is equal to

View answer and explanation
Correct option: B
−  4π6- \,\,\frac{{4\mathrm{\pi }}}{6}
Option A (− 4π6- \,\frac{{4\mathrm{\pi }}}{{6}}) is correct because it satisfies the given equation sin⁡θ=3cos⁡θ\sin \theta = \sqrt{3} \cos \theta and the condition π<θ<0\mathrm{\pi } < \mathrm{\theta } < 0.
103 Easy
103 of 261
Analyzing +4 | -1
The solution of the equation 4sin⁡4x+cos⁡4x=14{\sin ^4}x + {\cos ^4}x = 1 is
View answer and explanation
Correct option: D
None of the above
Option D (None of the above) is correct because the solution to the equation 4sin⁡4x+cos⁡4x=14{\sin ^4}x + {\cos ^4}x = 1 is x=nπx = n\pi, which is not represented by the other options.
104 Easy
104 of 261
Application +4 | -1
The maximum and minimum values of 7cos⁡θ+24sin⁡θ7 \cos \theta+24 \sin \theta are
View answer and explanation
Correct option: A
25 and −25
Option A (25 and -25) is correct because the maximum and minimum values of acos⁡θ+bsin⁡θa \cos \theta + b \sin \theta are a2+b2\sqrt{{a^2 + b^2}} and −a2+b2-\sqrt{{a^2 + b^2}} respectively.
105 Easy
105 of 261
Analyzing +4 | -1
The number of solutions of the pair of equations 2sin⁡2θ−cos⁡  2θ=02{\sin ^2}\theta - \cos \,\,2\theta = 0 and 2cos⁡2θ−3sin⁡  θ=02{\cos ^2}\theta - 3\sin \,\,\theta = 0 in the interval [0,   2π]\left[ {0,\,\;2\pi } \right] is

View answer and explanation
Correct option: C
Two
Option C (Two) is correct because solving the given equations yields sin⁡θ=12\sin \theta = \frac{1}{2}, which has two solutions in the interval [0,2π][0, 2\pi].

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106 Easy
106 of 261
Application +4 | -1
If 5  cos⁡  2  θ+2  cos⁡2  θ2+1=0, −π<θ<π5\,\,\cos \,\,2\;\theta + 2\,\,{\cos ^2}\,\,\frac{\theta }{2} + 1 = 0,\, - \pi < \theta < \pi, then θ=\theta =
View answer and explanation
Correct option: D
π3,  π−cos⁡−1  (3/5)\frac{\pi }{3},\,\,\pi - {\cos ^{ - 1}}\,\,\left( {3/5} \right)
Option D (π3, π−cos⁡−1  (3/5)\frac{\pi }{3},\,\pi - {\cos ^{ - 1}}\,\,\left( {3/5} \right)) is correct because these are the solutions to the given trigonometric equation within the specified interval.
107 Easy
107 of 261
Application +4 | -1
If 2  sin⁡2θ+3  cos⁡  θ+1=02\,\,{\sin ^2}\theta + \sqrt 3 \,\,\cos \,\,\theta + 1 = 0, then the value of θ\theta is
View answer and explanation
Correct option: C
5 π6\frac{{5\,\pi }}{6}
Option C is correct because it is the solution to the given trigonometric equation within the possible range of values.
108 Easy
108 of 261
Analyzing +4 | -1
If (sec⁡  θ−1)=(2−1)tan⁡  θ\left( {\sec \,\,\theta - 1} \right) = \left( {\sqrt 2 - 1} \right)\tan \,\,\theta, then θ=\theta =
View answer and explanation
Correct option: B
2  n  π,  2  n  π+π4,  n∈Z2\;n\;\pi ,\,\,2\;n\;\pi + \frac{\pi }{4},\,\,n \in Z
Option B (2  n  π, 2  n  π+π4, n∈Z2\;n\;\pi ,\,2\;n\;\pi + \frac{\pi }{4},\,n \in Z) is correct because it represents the general solutions obtained by solving the given trigonometric equation.
109 Easy
109 of 261
Analyzing +4 | -1
The minimum value of the expression sin⁡α+sin⁡β+sin⁡γ\sin \alpha+\sin \beta+\sin \gamma, where α,β\alpha, \beta and γ\gamma are real numbers satisfying α+β+γ=π\alpha+\beta+\gamma=\pi, is
View answer and explanation
Correct option: C
Negative
Option C (Negative) is correct because the minimum value of the expression can be negative when α, β, and γ are real numbers satisfying α+β+γ=π.
110 Easy
110 of 261
Analyzing +4 | -1
The equation sin⁡  x  cos⁡  x=2\sin \,\,x\,\,\cos \,\,x = 2 has
View answer and explanation
Correct option: D
No solution
Option D (No solution) is correct because the equation simplifies to sin⁡2x=4\sin 2x = 4, which is impossible since the range of the sine function is [-1, 1].
111 Easy
111 of 261
Application +4 | -1
The value of xx in (0,   π2)\left( {0,\;\,\frac{\pi }{2}} \right) satisfying the equation sin⁡ x  cos   x=14\sin \,x\,\,cos\,\;x = \frac{1}{4} is
View answer and explanation
Correct option: D
π12\frac{\pi }{{12}}
Option D (π12\frac{\pi }{{12}}) is correct because by using the double angle identity for sine and solving the resulting equation, we find that x=π12x = \frac{\pi }{{12}} is the solution in the given interval.
112 Easy
112 of 261
Application +4 | -1
sin⁡47∘+sin⁡61∘−sin⁡11∘−sin⁡25∘\sin 47^{\circ}+\sin 61^{\circ}-\sin 11^{\circ}-\sin 25^{\circ} is equal to
View answer and explanation
Correct option: D
cos⁡7∘\cos 7^{\circ}
Option D (cos⁡7∘\cos 7^{\circ}) is correct because the expression simplifies to cos⁡7∘\cos 7^{\circ} using trigonometric identities.
113 Easy
113 of 261
Application +4 | -1
If tan⁡  θ+tan⁡  2θ+3  tan⁡  θ  tan⁡  2θ=3\tan \,\,\mathrm{\theta } + \tan \,\,2\mathrm{\theta } + \sqrt 3 \,\,\tan \,\,\mathrm{\theta }\,\,\tan \,\,2\mathrm{\theta } = \sqrt 3, then
View answer and explanation
Correct option: C
θ=(3 n+1)π9,  ∀  n∈I\mathrm{\theta } = \frac{{\left( {3\,n + 1} \right)\pi }}{9},\,\,\forall \;n \in I
Option D (θ=(3n+1)π9,,∀  n∈I\mathrm{\theta } = \frac{{(3n + 1)\pi }}{9},\\,\forall \;n \in I) is correct because it is derived from solving the trigonometric equation using the tangent addition formula.
114 Easy
114 of 261
Application +4 | -1
If sin⁡θ\sin \theta and cos⁡θ\cos \theta are the roots of the equation ax2−bx+c=0a x^2-b x+c=0, then a,ba, b and cc satisfy the relation
View answer and explanation
Correct option: B
a2−b2+2ac=0a^2-b^2+2 a c=0
Option B (a2−b2+2ac=0a^2-b^2+2 a c=0) is correct because it is the relation derived from the given roots sin⁡θ\sin \theta and cos⁡θ\cos \theta of the quadratic equation.
115 Easy
115 of 261
Analyzing +4 | -1
The values of xx between   0\mathrm{\;0} and 2 π2\,\pi which satisfy the equation sin⁡  x 8  cos⁡2  x=1\sin \,\,x\,\sqrt {8\,\,{{\cos }^2}\,\,x} = 1 are in AP. The common difference of the AP is
View answer and explanation
Correct option: B
π4\frac{\mathrm{\pi }}{4}
Option B (π4\frac{\mathrm{\pi }}{4}) is correct because the values of x that satisfy the equation form an arithmetic progression with a common difference of π4\frac{\mathrm{\pi }}{4}.
116 Easy
116 of 261
Application +4 | -1
If tan⁡θ=n\tan \theta=\sqrt{n} for some non-square natural number nn, then sec⁡2θ\sec 2 \theta is
View answer and explanation
Correct option: A
A rational number
Option A (A rational number) is correct because sec⁡2θ\sec 2\theta can be expressed as a ratio of integers when tan⁡θ=n\tan \theta = \sqrt{n} and nn is a non-square natural number.
117 Easy
117 of 261
Analyzing +4 | -1
The value of (1+cos⁡π8)(1+cos⁡3π8)\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right) (1+cos⁡5π8)(1+cos⁡7π8)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right) is
View answer and explanation
Correct option: C
18\frac{1}{8}
Option C (18\frac{1}{8}) is correct because using trigonometric identities, the expression simplifies to 18\frac{1}{8}.
118 Easy
118 of 261
Application +4 | -1
If 0≤x≤π0 \le x \le \pi and 81sin⁡2x+  81cos⁡2x=30{81^{{{\sin }^2}x}} + \,\,{81^{{{\cos }^2}x}} = 30, then xx is equal to
View answer and explanation
Correct option: A
π6\frac{\pi }{6}
Option A (π6\frac{\pi }{6}) is correct because it satisfies the given trigonometric equation after simplification and solving for x.
119 Easy
119 of 261
Application +4 | -1
The value of cot⁡54∘tan⁡36∘+tan⁡20∘cot⁡70∘\frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot 70^{\circ}} is
View answer and explanation
Correct option: B
2
Option B (2) is correct because the expression simplifies to 1 + 1 after applying the complementary angle identities.
120 Easy
120 of 261
Application +4 | -1
If sec⁡2θ=2(1−tan⁡2θ){\sec ^2}\theta = \sqrt 2 \left( {1 - {{\tan }^2}\theta } \right), then θ=\theta =
View answer and explanation
Correct option: C
n  π±π8,  n∈Zn\;\pi \pm \frac{\pi }{8},\,\,n \in Z
Option C (n  π±π8, n∈Zn\;\pi \pm \frac{\pi }{8},\,n \in Z) is correct because it is derived by simplifying the given equation using trigonometric identities and solving for θ\theta.
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