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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Mixed difficulty

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361 Medium
361 of 581
Understanding +4 | -1
The least value of 3sin⁡2θ+4cos⁡2θ3 \sin ^2 \theta+4 \cos ^2 \theta is
View answer and explanation
Correct option: B
3
The correct answer is 3 because the minimum value of cos⁡2θ\cos^2 \theta is 0, making the minimum value of the expression 3.
362 Medium
362 of 581
Analyzing +4 | -1
The number of distinct solution of sin⁡5θcos⁡3θ=sin⁡9θcos⁡7θ\sin 5 \theta \cos 3 \theta=\sin 9 \theta \cos 7 \theta in [0,π2]\left[0, \frac{\pi}{2}\right], is
View answer and explanation
Correct option: D
9
Option D (9) is correct because there are nine distinct solutions for θ\theta in the given interval after simplifying the trigonometric equation.
363 Easy
363 of 581
Analyzing +4 | -1
If   tan⁡  x+cot⁡  x=2\;\tan \,\,x + \cot \,\,x = 2 then sin⁡2n  x+cos⁡2n  x  {\sin ^{2n}}\,\,x + {\cos ^{2n}}\,\,x\; is equal to
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the expression simplifies to 12n−1\frac{1}{2^{n-1}}, which is not among the given options.
364 Medium
364 of 581
Application +4 | -1
If 5(tan⁡2x−cos⁡2x)=2cos⁡2x+95\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9, then the value of cos⁡ 4x\cos \,4x is
View answer and explanation
Correct option: D
−79- \frac{7}{9}
Option D (−79- \frac{7}{9}) is correct because it is the value of cos⁡4x\cos 4x calculated from the given trigonometric equation.
365 Easy
365 of 581
Application +4 | -1
The value of 6(sin⁡6θ+cos⁡6θ)−9(sin⁡4θ+cos⁡4θ)+46\left(\sin ^6 \theta+\cos ^6 \theta\right)-9\left(\sin ^4 \theta+\cos ^4 \theta\right)+4 is
View answer and explanation
Correct option: C
1
The correct answer is 1 because the expression simplifies to 1 using trigonometric identities and algebraic manipulation.
366 Easy
366 of 581
Application +4 | -1
If sin⁡θ+cos⁡θ=0\sin \theta+\cos \theta=0 and 0<θ<π0 < \theta < \pi, then θ\theta is equal to
View answer and explanation
Correct option: D
3π4\frac{3 \pi}{4}
Option D (3π4\frac{3 \pi}{4}) is correct because it satisfies the equation sin⁡θ+cos⁡θ=0\sin \theta + \cos \theta = 0 within the given range 0<θ<π0 < \theta < \pi.
367 Medium
367 of 581
Analyzing +4 | -1
cos⁡2π7+cos⁡4π7+cos⁡6π7\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}
View answer and explanation
Correct option: C
Is a negative number
Option C (Is a negative number) is correct because the sum of the given cosines is equal to -1/2, which is a negative number.
368 Medium
368 of 581
Application +4 | -1
Thesumof all solutions of the equation
cos⁡x⋅cos⁡(π3+x)⋅cos⁡(π3−x)=14,x∈[0,6π]\cos x \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \cos \left(\frac{\pi}{3}-x\right)=\frac{1}{4}, x \in[0,6 \pi] is
View answer and explanation
Correct option: B
30π30 \pi
Option B (30π30 \pi) is correct because it is the sum of all solutions of the given equation in the specified interval.
369 Easy
369 of 581
Application +4 | -1
If tan⁡2  θ−(1+3)tan⁡  θ+3=0{\tan ^2}\,\,\mathrm{\theta } - \left( {1 + \sqrt 3 } \right)\tan \,\,\mathrm{\theta } + \sqrt 3 = 0, then the general value of θ\mathrm{\theta } is
View answer and explanation
Correct option: A
nπ  +  π4,   nπ  +  π3n\pi \,\, + \,\,\frac{\pi }{4},\;\,n\pi \,\, + \,\,\frac{\pi }{3}
Option A (nπ;+;π4,;nπ;+;π3n\pi \\; + \\; \frac{\pi }{{4}},\\;n\pi \\; + \\; \frac{\pi }{{3}}) is correct because it represents the general solutions obtained by solving the quadratic equation in tan⁡θ\tan \theta.
370 Easy
370 of 581
Application +4 | -1
The general value of θ\theta satisfying the equation 2  sin⁡2  θ−3  sin⁡  θ−2=02\,\,{\sin ^2}\,\,\theta - 3\,\,\sin \,\,\theta - 2 = 0 is
View answer and explanation
Correct option: A
nπ+(−1)n+1  π6n\pi + {\left( { - 1} \right)^{n + 1}}\,\,\frac{\pi }{6}
Option A (nπ+(−1)n+1 π6n\pi + {\left( { - 1} \right)^{n + 1}}\,\frac{\pi }{6}) is correct because it represents the general solution to the trigonometric equation 2sin⁡2 θ−3 sin⁡ θ−2=02{\sin ^2}\,\theta - 3\,\sin \,\theta - 2 = 0.
371 Medium
371 of 581
Analyzing +4 | -1
The range of 14∣sin⁡x∣−3\frac{1}{4|\sin x|-3} will be
View answer and explanation
Correct option: B
(−∞,−13]∪[1,∞)\left(-\infty,-\frac{1}{3}\right] \cup[1, \infty)
Option B ((-∞, -1/3] ∪ [1, ∞)) is correct because it represents the range of the function considering the bounds of the sine function.
372 Easy
372 of 581
Application +4 | -1
The number of solutions of cos⁡  2θ=sin⁡θ\cos \,\,2\theta = \sin \theta in (0,   2π)\left( {0,\,\;2\pi } \right) is
View answer and explanation
Correct option: C
3
Option C (3) is correct because there are three solutions to the equation cos⁡2θ=sin⁡θ\cos 2\theta = \sin \theta in the interval (0,2π)(0, 2\pi).
373 Easy
373 of 581
Application +4 | -1
The general solution of the equation 2sin⁡2x+3cos⁡x+1=02 \sin ^2 x+\sqrt{3} \cos x+1=0 is
View answer and explanation
Correct option: C
x=2nπ±5π6x=2 n \pi \pm \frac{5 \pi}{6}
Option C (x=2nπ±5π6x=2 n \pi \pm \frac{{5 \pi}}{{6}}) is the correct general solution obtained by solving the trigonometric equation for cos(x).
374 Easy
374 of 581
Analyzing +4 | -1
If sin⁡θ=x+ax,∀ x∈R∼{0}\sin \theta=x+\frac{a}{x}, \forall \, x \in R \sim\{0\}, then
View answer and explanation
Correct option: C
a≤14a \leq \frac{1}{4}
Option C (a≤14a \leq \frac{1}{4}) is correct because it is derived from the condition for the existence of real roots of the quadratic equation.
375 Easy
375 of 581
Application +4 | -1
The solution of the inequality −8≤5x−3<7-8 \leq 5 x-3 < 7 is [−a,b).[-a, b). Sum of &#39;aa&#39; and &#39;bb&#39; is
View answer and explanation
Correct option: A
3
Option A (3) is correct because solving the inequality −8≤5x−3<7-8 \leq 5x - 3 < 7 results in −1≤x<2-1 \leq x < 2, and the sum of the absolute values of the endpoints is 3.

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376 Easy
376 of 581
Analyzing +4 | -1
If A+B=π3A+B=\frac{\pi}{3}, where A,B∈R+A, B \in R^{+}, then the minimum value of sec⁡A+sec⁡B\sec A+\sec B is
View answer and explanation
Correct option: B
43\frac{4}{\sqrt{3}}
Option B (43\frac{4}{\sqrt{3}}) is correct because the minimum value of sec⁡A+sec⁡B\sec A + \sec B is found using the property that the average of secants is greater than or equal to the secant of the average angle.
377 Medium
377 of 581
Analyzing +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: A
(−π10,3π10)\left(-\frac{\pi}{10}, \frac{3 \pi}{10}\right)
Option A, (−π10,3π10)\left(-\frac{{\pi}}{{10}}, \frac{{3 \pi}}{{10}}\right), is correct because it represents the interval of x values that satisfy the given inequality within the specified range.
378 Easy
378 of 581
Application +4 | -1
{x∈R  :cos⁡  2x+2  cos⁡2x=2}\left\{ {x \in R\;:\cos \,\,2x + 2\,\,{{\cos }^2}x = 2} \right\} is equal to

View answer and explanation
Correct option: B
{nπ  ±  π6  :n∈Z}\left\{ {n\pi \,\, \pm \,\,\frac{\pi }{6}\;:n \in Z} \right\}
Option D (3/4) is correct because by using trigonometric identities and simplifying the expression, we can find the value to be 3/4.
379 Medium
379 of 581
Application +4 | -1
If 3sin⁡θ+5cos⁡θ=53 \sin \theta+5 \cos \theta=5, then the value of 5sin⁡θ−3cos⁡θ5 \sin \theta-3 \cos \theta is
View answer and explanation
Correct option: B
3
Option B (3) is correct because by using the given equation and trigonometric identities, we can find the value of the expression.
380 Medium
380 of 581
Application +4 | -1
If tan⁡2x+sec⁡x−a=0\tan ^2 x+\sec x-a=0 has atleast one solution, then the complete set of values of aa is
View answer and explanation
Correct option: D
[−1,∞)[-1, \infty)
Option D ([−1,∞)\left[-1, \infty\right)) is correct because by converting the equation to a quadratic in sec⁡x\sec x and analyzing the possible values of sec⁡x\sec x, we find that a≥−1a \geq -1 for the equation to have at least one solution.
381 Medium
381 of 581
Application +4 | -1
In a △ABC,cos⁡A+cos⁡B+cos⁡C\triangle A B C, \cos A+\cos B+\cos C is equal to
View answer and explanation
Correct option: D
1+rR1+\frac{r}{R}
Option D (1+rR1+\frac{r}{R}) is correct because by using trigonometric identities and the relationship between inradius, circumradius, and angles, the expression simplifies to 1+rR1+\frac{r}{R}.
382 Easy
382 of 581
Analyzing +4 | -1
The number of solutions of the equation tan⁡  θ+sec⁡  θ=2  cos⁡  θ\tan \,\,\theta + \sec \,\,\theta = 2\,\,\cos \,\,\theta lying in the interval [0,   2  π]\left[ {0,\;\,2\;\pi } \right], is

View answer and explanation
Correct option: C
2\mathrm{2}
Option C (2) is correct because solving the equation tan⁡θ+sec⁡θ=2cos⁡θ\tan \theta + \sec \theta = 2 \cos \theta yields two valid solutions within the interval [0,2π][0, 2\pi].
383 Easy
383 of 581
Application +4 | -1
If 2cos⁡θ−sin⁡θ+2=02 \cos \theta-\sin \theta+2=0, then the value of 10cos⁡θ+15sin⁡θ10 \cos \theta+15 \sin \theta is equal to:
View answer and explanation
Correct option: B
6
Option B (6) is correct because by solving the given trigonometric equation and substituting the values of cos⁡θ\cos \theta and sin⁡θ\sin \theta into the expression 10cos⁡θ+15sin⁡θ10 \cos \theta + 15 \sin \theta, we get 6.
384 Easy
384 of 581
Application +4 | -1
The solution set of the inequality 4x+3<6x+74 x+3 < 6 x+7 is (−a,∞)(-a, \infty). The value of &#39;aa&#39; is
View answer and explanation
Correct option: A
2
Option A (2) is correct because solving the inequality 4x+3<6x+74x + 3 < 6x + 7 results in x>−2x > -2, which corresponds to the interval (−2,∞)(-2, \infty).
385 Medium
385 of 581
Analyzing +4 | -1
If a,b∈[0,π]a, b \in[0, \pi] and the equation x2+4+3sin⁡(ax+b)−2x=0x^2+4+3 \sin (a x+b)-2 x=0 has atleast one solution, then the value of (a+b)(a+b) can be
View answer and explanation
Correct option: B
3π2\frac{3 \pi}{2}
Option D ((n+1)24n\frac{{(n+1)^2}}{{4 n}}) is correct because it represents the maximum value of sec⁡2(x−y)\sec ^2(x-y) derived by using the AM-GM inequality after expressing sec⁡2(x−y)\sec^2(x-y) in terms of tan⁡y\tan y.
386 Medium
386 of 581
Analyzing +4 | -1
If log⁡cos⁡xtan⁡x+log⁡sin⁡xcot⁡x=0\log _{\cos x} \tan x+\log _{\sin x} \cot x=0, then the most general solution of xx is
View answer and explanation
Correct option: B
2nπ+π4,n∈Z2 n \pi+\frac{\pi}{4}, n \in Z
Option B (2nπ+π4,n∈Z2 n \pi+\frac{\pi}{4}, n \in Z) is correct because it represents the general solution of the equation, considering the domains of the logarithmic and trigonometric functions involved.
387 Medium
387 of 581
Application +4 | -1
If 4cos⁡θ+5sin⁡θ=14 \cos \theta+5 \sin \theta=1 and α\alpha satisfy equation and α∈(−π2,π2)\alpha \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) then, value of tan⁡α\tan \alpha is
View answer and explanation
Correct option: B
10−1012\frac{\sqrt{10}-10}{12}
The correct answer is B; 10−1012\frac{{\sqrt{{10}} - 10}}{{12}}. It involves using trigonometric identities to convert the given equation into a quadratic equation in terms of tan⁡(θ/2)\tan(\theta/2), solving for tan⁡(θ/2)\tan(\theta/2), and then finding tan⁡(α)\tan(\alpha).
388 Easy
388 of 581
Analyzing +4 | -1
If equation sin⁡4x=1+tan⁡8x\sin ^4 x=1+\tan ^8 x, then xx is equal to
View answer and explanation
Correct option: A
ϕ\phi
Option A (ϕ\phi) is correct because the equation sin⁡4x=1+tan⁡8x\sin^4 x = 1 + \tan^8 x has no solution since sin⁡4x\sin^4 x cannot be 1 and tan⁡8x\tan^8 x be 0 simultaneously.
389 Medium
389 of 581
Analyzing +4 | -1
If 2sin⁡2((π2)cos⁡2x)=1−cos⁡(πsin⁡2x),x≠(2n+1)π2,n∈I2 \sin ^2\left(\left(\frac{\pi}{2}\right) \cos ^2 x\right)=1-\cos (\pi \sin 2 x), x \neq(2 n+1) \frac{\pi}{2}, n \in I, then cos⁡2x\cos 2 x is equal to
View answer and explanation
Correct option: B
35\frac{3}{5}
Option B (35\frac{3}{5}) is correct because it is derived by simplifying the given trigonometric equation and solving for cos⁡2x\cos 2x.
390 Easy
390 of 581
Analyzing +4 | -1
The number of solutions of the equation 1+sin⁡x  sin⁡2x2=01 + \sin x\,\,{\sin ^2}\frac{x}{2} = 0 in [− π, π]\left[ { - \,\mathrm{\pi },\,\mathrm{\pi }} \right] is
View answer and explanation
Correct option: A
zero
Option A (zero) is correct because the equation 1+sin⁡xsin⁡2x2=01 + \sin x \sin^2 \frac{x}{2} = 0 simplifies to sin⁡2x−2sin⁡x=4\sin 2x - 2 \sin x = 4, which has no solutions since the range of sin⁡\sin is [-1, 1].
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