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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Medium

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Showing 31-60 of 294 questions
31 Medium
31 of 294
Analyzing +4 | -1
The number of solutions of sin⁡3x=cos⁡2x\sin 3 x=\cos 2 x, in the interval (π2,π)\left(\frac{\pi}{2}, \pi\right) is
View answer and explanation
Correct option: D
1
Option D (1) is correct because after simplifying the equation sin⁡3x=cos⁡2x\sin 3x = \cos 2x and considering the interval (π2,π)(\frac{\pi}{2}, \pi), there is only one solution.
32 Medium
32 of 294
Analyzing +4 | -1
If xx and yy are acute angles such that cos⁡x+cos⁡y=32\cos x+\cos y=\frac{3}{2} and sin⁡x+sin⁡y=34\sin x+\sin y=\frac{3}{4}, then sin⁡(x+y)\sin (x+y) equals
View answer and explanation
Correct option: D
45\frac{4}{5}
Option D (45\frac{4}{5}) is correct because it is the value of sin⁡(x+y)\sin(x+y) calculated using the given equations and trigonometric identities.
33 Medium
33 of 294
Analyzing +4 | -1
If asin⁡x+bcos⁡(x+θ)+bcos⁡(x−θ)=da \sin x+b \cos (x+\theta)+b \cos (x-\theta)=d, then the minimum value of ∣cos⁡θ∣|\cos \theta| is
View answer and explanation
Correct option: A
12∣b∣d2−a2\frac{1}{2|b|} \sqrt{d^2-a^2}
Option A (12∣b∣d2−a2\frac{1}{2|b|} \sqrt{d^2-a^2}) is correct because it represents the minimum value of |cos θ| derived from the given equation.
34 Medium
34 of 294
Understanding +4 | -1
Solution of the inequality sin⁡x>1/2\sin x > 1 / \sqrt{2} belongs to
View answer and explanation
Correct option: A
(2nπ+π4,2nπ+3π4)\left(2 n \pi+\frac{\pi}{4}, 2 n \pi+\frac{3 \pi}{4}\right) where n∈Zn \in \mathbb{Z}
Option A ((2nπ+π4,2nπ+3π4)\left(2 n \pi+\frac{\pi}{4}, 2 n \pi+\frac{3 \pi}{4}\right) where n∈Zn \in \mathbb{{Z}}) is correct because sin⁡x>1/2\sin x > 1/\sqrt{{2}} when xx is between π/4\pi/4 and 3π/43\pi/4 in each period.
35 Medium
35 of 294
Analyzing +4 | -1
The number of solutions of 2sin⁡x+cos⁡x=32 \sin x+\cos x=3 is
View answer and explanation
Correct option: D
No solution
Option D (No solution) is correct because the maximum value of 2sin⁡x+cos⁡x2 \sin x + \cos x is 5\sqrt{5}, which is less than 3.
36 Medium
36 of 294
Analyzing +4 | -1
The total number of ordered pairs (r,  θ)\left( {r,\,\,\theta } \right) satisfying r  sin⁡  θ=3,  r=4(1+sin⁡  θ)r\,\,\sin \,\,\theta = 3,\,\,r = 4\left( {1 + \sin \,\,\theta } \right), where r>0r > 0 and θ∈[−π,  π]\theta \in \left[ { - \pi ,\,\,\pi } \right] is
View answer and explanation
Correct option: B
2\mathrm{2}
Option B (2) is correct because solving the system of equations yields two valid solutions for θ\theta within the given interval.
37 Medium
37 of 294
Analyzing +4 | -1
The number of roots of the equation ∑k=17sin⁡(kx)=7sin⁡4x\sum\limits_{{k=1}}^7 \sin (k x)=7 \sin 4 x in the interval [0,2π][0,2 \pi] is
View answer and explanation
Correct option: D
9
The correct answer is D (9) because the equation simplifies to sin(4x) = 0 or cos(x) = 1, leading to 9 solutions in the interval [0, 2π].
38 Medium
38 of 294
Analyzing +4 | -1
If x1x_1 and x2x_2 are two positive values of xx for which 2cos⁡x,∣cos⁡x∣2 \cos x,|\cos x| and 3sin⁡2x−23 \sin ^2 x-2 are in GP. The minimum value of ∣x1−x2∣\left|x_1-x_2\right| is
View answer and explanation
Correct option: C
2cos⁡−1(23)2 \cos ^{-1}\left(\frac{2}{3}\right)
Option C (2cos⁡−1(23)2 \cos ^{-1}\left(\frac{2}{3}\right)) is correct because it represents the minimum difference between two positive values of x that satisfy the given geometric progression condition.
39 Medium
39 of 294
Application +4 | -1
The general solution of the equation tan⁡θ+tan⁡2θ+3tan⁡θtan⁡2θ=3\tan \theta+\tan 2 \theta+\sqrt{3} \tan \theta \tan 2 \theta=\sqrt{3} is
View answer and explanation
Correct option: A
θ=(3n+1)π9\theta=(3 n+1) \frac{\pi}{9}
Option A (θ=(3n+1)π9\theta=(3 n+1) \frac{{\pi}}{{9}}) is correct because it represents the general solution derived from solving the given trigonometric equation.
40 Medium
40 of 294
Application +4 | -1
If 3sin⁡α=5sin⁡β3 \sin \alpha=5 \sin \beta, then tan⁡α+β2tan⁡α−β2\frac{\tan \frac{\alpha+\beta}{2}}{\tan \frac{\alpha-\beta}{2}} is equal to
View answer and explanation
Correct option: D
4
Option D (4) is correct because by applying the componendo and dividendo rule and the sum-to-product trigonometric identities, the expression simplifies to 4.
41 Medium
41 of 294
Application +4 | -1
The most general solution of the equation
8  tan⁡2  θ2=1+sec⁡  θ,    is8\,\,{\tan ^2}\,\,\frac{\theta }{2} = 1 + \sec \,\,\theta ,\;\,\,\mathrm{is}
View answer and explanation
Correct option: A
θ=2n  π  ±  cos⁡−1  (13)\theta = 2n\;\pi \,\, \pm \,\,{\cos ^{ - 1}}\,\,\left( {\frac{1}{3}} \right)
Option A (θ=2n  π ± cos⁡−1 (13)\theta = 2n\;\pi \, \pm \,{\cos ^{ - 1}}\,\left( {\frac{1}{3}} \right)) is correct because it represents the general solution obtained by solving the given trigonometric equation.
42 Medium
42 of 294
Application +4 | -1
The equation kcos⁡x−3sin⁡x=k+1k \cos x-3 \sin x=k+1 is solvable only if
View answer and explanation
Correct option: B
k∈(−∞,4]k \in(-\infty, 4]
Option B (k∈(−∞,4]k \in(-\infty, 4]) is correct because the equation is solvable only when the absolute value of (k+1) is less than or equal to the square root of (k2 + 9), leading to k <= 4.
43 Medium
43 of 294
Analyzing +4 | -1
If sum of all the solutions of the equation 8cos⁡x(cos⁡(π6+x)cos⁡(π6−x)−12)=18 \cos x\left(\cos \left(\frac{\pi}{6}+x\right) \cos \left(\frac{\pi}{6}-x\right)-\frac{1}{2}\right)=1 in [0,π][0, \pi] is kπ{k} \pi, then k{k} is equal to:
View answer and explanation
Correct option: C
139\frac{13}{9}
The value of k is 13/9. This is found by simplifying the trigonometric equation, solving for x, and summing the solutions in the given interval.
44 Medium
44 of 294
Analyzing +4 | -1
If cos⁡A=mcos⁡B\cos A=m \cos B and cot⁡(A+B2)=λ/tan⁡(B−A2)\cot \left(\frac{A+B}{2}\right)=\lambda/\tan \left(\frac{B-A}{2}\right), then λ\lambda is equal to
View answer and explanation
Correct option: C
m+1m−1\frac{m+1}{m-1}
Option C (m+1m−1\frac{{m+1}}{{m-1}}) is correct because it's derived by applying the componendo and dividendo rule and trigonometric identities to the given equation.
45 Medium
45 of 294
Understanding +4 | -1
If sin⁡θ=12\sin \theta=\frac{1}{2} and tan⁡θ=13,∀ n∈I\tan \theta=\frac{1}{\sqrt{3}}, \forall \, n \in I, then most general value of θ\theta is
View answer and explanation
Correct option: A
2nπ+π6,∀ n∈I2 n \pi+\frac{\pi}{6}, \forall \, n \in I
The correct answer is 2nπ+π6,∀,n∈I2 n \pi+\frac{{\pi}}{{6}}, \forall \\, n \in I because it represents the general solution that satisfies both sin⁡θ=12\sin \theta = \frac{{1}}{{2}} and tan⁡θ=13\tan \theta = \frac{{1}}{{\sqrt{{3}}}} .

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46 Medium
46 of 294
Analyzing +4 | -1
Number of ordered pairs (a,x)(a, x) satisfying the equation sec⁡2(a+2)x+a2−1=0,−π<x<π\sec ^2(a+2) x+a^2-1=0,-\pi < x < \pi is
View answer and explanation
Correct option: C
3
Option C (3) is correct because solving the given equation leads to three ordered pairs that satisfy the conditions.
47 Medium
47 of 294
Analyzing +4 | -1
If S=cos⁡2πn+cos⁡22πn+…+cos⁡2(n−1)πnS=\cos ^2 \frac{\pi}{n}+\cos ^2 \frac{2 \pi}{n}+\ldots+\cos ^2 \frac{(n-1) \pi}{n}, then SS equals
View answer and explanation
Correct option: C
12(n−2)\frac{1}{2}(n-2)
Option C (12(n−2)\frac{1}{2}(n-2)) is correct because the sum simplifies to this expression using trigonometric identities.
48 Medium
48 of 294
Analyzing +4 | -1
If 2sin⁡α1+cos⁡α+sin⁡α=x\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}=x, then 1−cos⁡α−sin⁡αcos⁡α\frac{1-\cos \alpha-\sin \alpha}{\cos \alpha} is equal to
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because simplifying the given expression leads to 1−cos⁡α−sin⁡αcos⁡α=−x\frac{{1 - \cos \alpha - \sin \alpha}}{{\cos \alpha}} = -x, which is not among the provided options.
49 Medium
49 of 294
Application +4 | -1
The set of all x∈(−π,π)x \in(-\pi, \pi) satisfying ∣4sin⁡x−1∣<5|4 \sin x-1| < \sqrt{5} is given by
View answer and explanation
Correct option: A
(−π10,3π10)\left(-\frac{\pi}{10}, \frac{3 \pi}{10}\right)
Option A is correct because it represents the interval where the inequality ∣4sin⁡x−1∣<5|4 \sin x - 1| < \sqrt{5} holds true within the specified range of (−π,π)(-\pi, \pi).
50 Medium
50 of 294
Analyzing +4 | -1
If ∣sin⁡xcos⁡xcos⁡xcos⁡xsin⁡xcos⁡xcos⁡xcos⁡xsin⁡x∣=0\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0, then the number of distinct real roots of this equation in the interval −π/2<x<π/2-\pi / 2 < x < \pi / 2 is
View answer and explanation
Correct option: C
2
Option C (2) is correct because solving the determinant equation leads to two distinct real roots within the given interval.
51 Medium
51 of 294
Analyzing +4 | -1
If sin⁡  (π  cot⁡  θ)=cos⁡(π  tan⁡  θ)\sin \,\,\left( {\pi \,\,\cot \,\,\theta } \right) = \cos \left( {\pi \,\,\tan \,\,\theta } \right), then
View answer and explanation
Correct option: A
cot⁡  2θ=±  14,   −34\cot \,\,2\theta = \pm \,\,\frac{1}{4},\,\; - \frac{3}{4}
Option A (cot⁡2θ=±14,−34\cot 2\theta = \pm \frac{1}{4}, -\frac{3}{4}) is correct because by using trigonometric identities and solving the equation, we find the possible values of cot⁡2θ\cot 2\theta.
52 Medium
52 of 294
Analyzing +4 | -1
If exp⁡[(sin⁡2x+sin⁡4x+sin⁡6x+…)ln⁡2]\exp ^{\left[\left(\sin ^2 x+\sin ^4 x+\sin ^6 x+\ldots\right) \ln 2\right]} satisfies the equation y2−9y+8=0y^2-9 y+8=0, then the value of cos⁡xcos⁡x+sin⁡x,0<x<π2\frac{\cos x}{\cos x+\sin x}, 0 < x < \frac{\pi}{2}, is
View answer and explanation
Correct option: B
3−12\frac{\sqrt{3}-1}{2}
Option B, 3−12\frac{{\sqrt{{3}}-1}}{{2}}, is correct because it is the simplified value of cos⁡xcos⁡x+sin⁡x\frac{{\cos x}}{{\cos x+\sin x}} when x=π3x = \frac{{\pi}}{{3}}, which is obtained from solving the given equation.
53 Medium
53 of 294
Application +4 | -1
From the identity sin⁡3x=3sin⁡x−4sin⁡3x\sin 3 x=3 \sin x-4 \sin ^3 x it follows that if xx is real and ∣x∣<1|x| < 1, then
View answer and explanation
Correct option: B
(3x−4x3)≤1\left(3 x-4 x^3\right) \leq 1
Option B ((3x−4x3)≤1(3 x-4 x^3) \leq 1) is correct because given ∣x∣<1|x| < 1, we can substitute x=sin⁡θx = \sin \theta, and using the identity sin⁡3θ=3sin⁡θ−4sin⁡3θ\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta, we know that sin⁡3θ\sin 3\theta is always between -1 and 1, hence less than or equal to 1.
54 Medium
54 of 294
Application +4 | -1
Total number of solutions of cos⁡xcos⁡2xcos⁡3x=14\cos x \cos 2 x \cos 3 x=\frac{1}{4} in [0,π][0, \pi] is
View answer and explanation
Correct option: B
6
Option B (6) is correct because by simplifying the given trigonometric equation, we find that there are 6 solutions in the interval [0, π\pi].
55 Medium
55 of 294
Analyzing +4 | -1
The number of roots of the equation x+2  tan⁡  x=π2x + 2\,\,\tan \,\,x = \frac{\pi }{2} in the interval [0,  2  π]\left[ {0,\,\,2\;\pi } \right], is
View answer and explanation
Correct option: C
3\mathrm{3}
Option C (3) is correct because the graphs of y=tan⁡xy = \tan x and y=π4−x2y = \frac{\pi}{4} - \frac{x}{2} intersect at three points in the given interval.
56 Medium
56 of 294
Analyzing +4 | -1
The equation esin⁡x−e−sin⁡x−4=0e^{\sin x}-e^{-\sin x}-4=0 has
View answer and explanation
Correct option: A
No solution
Option A (No solution) is correct because the equation leads to sin⁡x>1\sin x > 1, which is impossible for real values of x.
57 Medium
57 of 294
Application +4 | -1
The number of values of xx in the interval [0,   5π]\left[ {0,\;\,5\mathrm{\pi }} \right] satisfying the equation 3  sin⁡2x−7  sin⁡  x+2=03\,\,{\sin ^2}x - 7\,\,\sin \,\,x + 2 = 0 is
View answer and explanation
Correct option: C
6
Option D (6) is correct because solving the trigonometric equation and considering the interval [0, 5π\pi] yields 6 solutions.
58 Medium
58 of 294
Application +4 | -1
If tan⁡  θ,  cos⁡  θ,  16  sin⁡  θ\tan \,\,\theta ,\,\,\cos \,\,\theta ,\,\,\frac{1}{6}\,\,\sin \,\,\theta are in G.P., then general value of θ\theta is
View answer and explanation
Correct option: A
2  n   π  ±  π3, n∈Z2\;n\;\,\pi \,\, \pm \,\,\frac{\pi }{3},\,n \in Z
Option A (2  n  π,±,π3, n∈Z2\;n\;\pi \\, \pm \\, \frac{\pi }{3},\,n \in Z) is correct because solving the geometric progression condition leads to cos⁡θ=12\cos \theta = \frac{1}{2}, which gives the general solution θ=2nπ±π3\theta = 2n\pi \pm \frac{\pi}{3}.
59 Medium
59 of 294
Application +4 | -1
sin⁡2π7+sin⁡4π7+sin⁡8π7\sin \frac{2 \pi}{7}+\sin \frac{4 \pi}{7}+\sin \frac{8 \pi}{7} is equal to
View answer and explanation
Correct option: A
72\frac{\sqrt{7}}{2}
Option A (72\frac{{\sqrt{{7}}}}{{2}}) is correct because the sum of sines simplifies to 72\frac{{\sqrt{{7}}}}{{2}} using trigonometric identities and sum-to-product formulas.
60 Medium
60 of 294
Application +4 | -1
A value of θ\theta satisfying sin⁡5θ−sin⁡3θ+sin⁡θ=0\sin 5 \theta-\sin 3 \theta+\sin \theta=0 such that 0<θ<π20 < \theta < \frac{\pi}{2} is
View answer and explanation
Correct option: B
π6\frac{\pi}{6}
Option B (π6\frac{\pi}{6}) is the correct answer because it satisfies the given trigonometric equation within the specified range.
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