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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Mixed difficulty

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541 Medium
541 of 581
Analyzing +4 | -1
If x=α,βx=\alpha, \beta satisfies both the equation cos⁡2x+acos⁡x+b=0\cos ^2 x+a \cos x+b=0 and sin⁡2x+psin⁡x+q=0\sin ^2 x+p \sin x+q=0, then the relation between a,b,pa, b, p and qq is
View answer and explanation
Correct option: C
2(b+q)=a2+p2−22(b+q)=a^2+p^2-2
Option C (2(b+q)=a2+p2−22(b+q)=a^2+p^2-2) is correct because it represents the relationship between the coefficients derived from the given equations and trigonometric identities.
542 Medium
542 of 581
Analyzing +4 | -1
The equation tan⁡4x−2sec⁡2x+a2=0\tan ^4 x-2 \sec ^2 x+a^2=0 will have atleast one solution, if
View answer and explanation
Correct option: C
∣a∣≤3|a| \leq \sqrt{3}
Option C (∣a∣≤3|a| \leq \sqrt{3}) is correct because the equation has at least one solution when the discriminant of the transformed equation is non-negative.
543 Medium
543 of 581
Analyzing +4 | -1
In a △ABC\triangle A B C, the maximum value of acos⁡2A2+bcos⁡2B2+ccos⁡2C2a+b+c\frac{a \cos ^2 \frac{A}{2}+b \cos ^2 \frac{B}{2}+c \cos ^2 \frac{C}{2}}{a+b+c} is
View answer and explanation
Correct option: C
34\frac{3}{4}
Option C (34\frac{3}{4}) is correct because the maximum value of the expression is 34\frac{3}{4} when using trigonometric identities and the inequality R≥2rR \geq 2r.
544 Easy
544 of 581
Application +4 | -1
Find the general solution of the equation sin⁡x−3sin⁡2x+sin⁡3x=cos⁡x−3cos⁡2x+cos⁡3x\sin x-3 \sin 2 x+\sin 3 x=\cos x-3 \cos 2 x+\cos 3 x
View answer and explanation
Correct option: C
nπ2+π8\frac{n \pi}{2}+\frac{\pi}{8}
Option C (nπ2+π8\frac{{n \pi}}{{2}}+\frac{{\pi}}{{8}}) is the correct general solution obtained by simplifying the given trigonometric equation and solving for x.
545 Easy
545 of 581
Application +4 | -1
The solution of the equation tan⁡θtan⁡2θ=1\tan \theta \tan 2 \theta=1 is
View answer and explanation
Correct option: D
nπ±π6n \pi \pm \frac{\pi}{6}
Option D (nπ±π6n \pi \pm \frac{{\pi}}{{6}}) is correct because it represents the general solution derived from solving the trigonometric equation tan⁡θtan⁡2θ=1\tan \theta \tan 2 \theta = 1.
546 Medium
546 of 581
Analyzing +4 | -1
The sum of the series ∑n=1∞sin⁡(n!  π720)\sum\limits_{n = 1}^\infty {\sin \left( {\frac{{n!\,\,\pi }}{{720}}} \right)} is
View answer and explanation
Correct option: C
sin⁡(π6)+sin⁡(π30)+sin⁡(π120)+sin⁡(π360)+sin⁡(π720)\sin \left(\frac{\pi}{6}\right)+\sin \left(\frac{\pi}{30}\right)+\sin \left(\frac{\pi}{120}\right)+\sin \left(\frac{\pi}{360}\right)+\sin \left(\frac{\pi}{720}\right)
Option C (sin⁡(π6)+sin⁡(π30)+sin⁡(π120)+sin⁡(π360)+sin⁡(π720)\sin \left(\frac{\pi}{6}\right)+\sin \left(\frac{\pi}{30}\right)+\sin \left(\frac{\pi}{120}\right)+\sin \left(\frac{\pi}{360}\right)+\sin \left(\frac{\pi}{720}\right)) is correct because after the first five terms, all subsequent terms become multiples of π\pi and thus their sine values are zero.
547 Medium
547 of 581
Application +4 | -1
If sin⁡x+3cos⁡x=2\sin x+\sqrt{3} \cos x=\sqrt{2}, then xx is
View answer and explanation
Correct option: B
2nπ+5π12,2nπ−π122 n \pi+\frac{5 \pi}{12}, 2 n \pi-\frac{\pi}{12}
Option B (2nπ+5π12,2nπ−π122 n \pi+\frac{{5 \pi}}{{12}}, 2 n \pi-\frac{{\pi}}{{12}}) is correct because it represents the general solution derived from solving the given trigonometric equation.
548 Medium
548 of 581
Analyzing +4 | -1
Consider the following statements :

1. cot⁡  θ−tan⁡  θ,    then    θ=(4n+1)  π8\cot \,\,\mathrm{\theta } - \tan \,\,\mathrm{\theta },\,\,\;\mathrm{then}\,\,\mathrm{\;\theta } = \left( {4n + 1} \right)\,\,\frac{\pi }{8}

2. sin⁡  2x  +  cos⁡  2x  +  sin⁡  x+cos⁡  x+1=0\sin \,\,2x\,\, + \,\,\cos \,\,2x\,\, + \,\,\sin \,\,x + \cos \,\,x + 1 = 0 has no solution in the Ist quadrant.
Which of these is/are correct?
View answer and explanation
Correct option: C
Both of these
Option C (Both of these) is correct because both statements regarding trigonometric identities and solutions are accurate.
549 Easy
549 of 581
Application +4 | -1
For m≠nm \ne n, if tan⁡  m  θ=tan⁡  n  θ\tan \,\,m\;\theta = \tan \,\,n\;\theta, the different values of θ\theta are in
View answer and explanation
Correct option: A
A.P.
Option A (A.P.) is the correct answer because the values of θ\theta form an arithmetic progression.
550 Easy
550 of 581
Analyzing +4 | -1
If x  cos⁡  θ=y  cos⁡(θ+2π3)=z  cos⁡(θ+4π3)x\,\,\cos \,\,\theta = y\,\,\cos \left( {\theta + \frac{{2\pi }}{3}} \right) = z\,\,\cos \left( {\theta + \frac{{4\pi }}{3}} \right), then the value of 1x+1y+1z\frac{1}{x} + \frac{1}{y} + \frac{1}{z} is equal to
View answer and explanation
Correct option: C
0
Option C (0) is correct because the sum of cosines with angles in arithmetic progression is zero when the common difference is 2π3\frac{{2\pi}}{3}.
551 Easy
551 of 581
Understanding +4 | -1
Which of the following is not correct?
View answer and explanation
Correct option: C
sec⁡θ=12\sec \theta=\frac{1}{2}
Option C (sec⁡θ=12\sec \theta=\frac{1}{2}) is correct because the range of the secant function is (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty), so it cannot be between -1 and 1.
552 Medium
552 of 581
Application +4 | -1
If tan⁡(πcos⁡x)=cot⁡(πsin⁡x)\tan (\pi \cos x)=\cot (\pi \sin x), then cos⁡(x−π4)\cos \left(x-\frac{\pi}{4}\right) is
View answer and explanation
Correct option: B
122\frac{1}{2 \sqrt{2}}
Option B (122\frac{{1}}{{2 \sqrt{{2}}}}) is correct because it is the value obtained after simplifying the given trigonometric equation and using the cosine subtraction formula.
553 Easy
553 of 581
Application +4 | -1
If cos⁡  3  x+sin⁡(2c−7  π6)=−2\cos \,\,3\;x + \sin \left( {2c - \frac{{7\;\pi }}{6}} \right) = - 2 then x=x =
View answer and explanation
Correct option: A
π3  (6k+1),  k∈Z\frac{\pi }{3}\,\,\left( {6k + 1} \right),\,\,k \in Z
Option A is correct because it provides the general solution for x that satisfies the given trigonometric equation.
554 Easy
554 of 581
Analyzing +4 | -1
If x,y∈[0,2π]x, y \in[0,2 \pi], then total number of ordered pairs (x,y)(x, y) satisfying sin⁡xcos⁡y=1\sin x \cos y=1 is
View answer and explanation
Correct option: B
3
Option B (3) is correct because there are three ordered pairs (x, y) that satisfy the given equation sin(x)cos(y) = 1 within the specified interval.
555 Medium
555 of 581
Application +4 | -1
The value of cos⁡π7cos⁡4π7cos⁡5π7\cos \frac{\pi}{7} \cos \frac{4 \pi}{7} \cos \frac{5 \pi}{7} is
View answer and explanation
Correct option: D
18\frac{1}{8}
The correct answer is 18\frac{1}{8} because the expression simplifies using trigonometric identities and the double angle formula.

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556 Hard
556 of 581
Application +4 | -1
In a triangle ABCA B C, the angle BB is greater than the angle AA. If the values of angles AA and BB satisfy the equation 3sin⁡x−4sin⁡3x−k=0,0<k<13 \sin x-4 \sin ^{3} x-k=0,0 < k < 1, then the values of CC is
View answer and explanation
Correct option: C
2π/32 \pi / 3
Option C is correct because it is derived from the sine triple angle identity and the given conditions, leading to the correct value for angle C.
557 Medium
557 of 581
Application +4 | -1
If sin⁡(α+β)=1\sin (\alpha+\beta)=1 and sin⁡(α−β)=12\sin (\alpha-\beta)=\frac{1}{2}, then tan⁡(α+2β)tan⁡(2α+β)\tan (\alpha+2 \beta) \tan (2 \alpha+\beta) is equal to
View answer and explanation
Correct option: A
1
Option A (1) is correct because solving the system of equations for α\alpha and β\beta and substituting into the expression yields a value of 1.
558 Medium
558 of 581
Application +4 | -1
The value of cos⁡2π15cos⁡4π15cos⁡8π15cos⁡16π15\cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{16 \pi}{15} is
View answer and explanation
Correct option: A
116\frac{1}{16}
Option A (116\frac{{1}}{{16}}) is correct because the given expression simplifies to 116\frac{{1}}{{16}} by repeatedly using the identity 2sin⁡xcos⁡x=sin⁡2x2 \sin x \cos x = \sin 2x.
559 Medium
559 of 581
Application +4 | -1
If α+β+γ=2π\alpha+\beta+\gamma=2 \pi, then
View answer and explanation
Correct option: A
tan⁡α2+tan⁡β2+tan⁡γ2=tan⁡α2tan⁡β2tan⁡γ2\tan \frac{\alpha}{2}+\tan \frac{\beta}{2}+\tan \frac{\gamma}{2}=\tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2}
Option A (tan⁡α2+tan⁡β2+tan⁡γ2=tan⁡α2tan⁡β2tan⁡γ2\tan \frac{\alpha}{2}+\tan \frac{\beta}{2}+\tan \frac{\gamma}{2}=\tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2}) is correct because it is derived from the given condition α+β+γ=2π\alpha+\beta+\gamma=2 \pi by using the tangent addition formula.
560 Medium
560 of 581
Analyzing +4 | -1
If cos⁡  x≠−12\cos \,\,x \ne - \frac{1}{2}, then the solutions of cos⁡  x+cos⁡  2x+cos⁡  3x=0\cos \,\,x + \cos \,\,2x + \cos \,\,3x = 0 are
View answer and explanation
Correct option: A
2nπ  ±  π4, n∈Z2n\pi \,\, \pm \,\,\frac{\pi }{4},\,n \in Z
Option A (2nπ ± π4, n∈Z2n\pi \, \pm \, \frac{\pi }{4},\,n \in Z) is correct because solving the trigonometric equation cos⁡x+cos⁡2x+cos⁡3x=0\cos x + \cos 2x + \cos 3x = 0 yields this general solution.
561 Medium
561 of 581
Application +4 | -1
Let cos⁡(α+β)=45\cos (\alpha+\beta)=\frac{4}{5} and sin⁡(α−β)=513\sin (\alpha-\beta)=\frac{5}{13}, where 0≤α,β≤π40 \leq \alpha, \beta \leq \frac{\pi}{4}. Then, tan⁡2α\tan 2 \alpha is equal to
View answer and explanation
Correct option: B
5633\frac{56}{33}
Option B (5633\frac{56}{33}) is correct because it is the value of tan⁡2α\tan 2\alpha calculated using the given trigonometric values and the tangent addition formula.
562 Easy
562 of 581
Application +4 | -1
The numerical value of tan⁡π3+2tan⁡2π3+4tan⁡4π3+8tan⁡8π3\tan \frac{\pi}{3}+2 \tan \frac{2 \pi}{3}+4 \tan \frac{4 \pi}{3}+8 \tan \frac{8 \pi}{3} is
View answer and explanation
Correct option: A
−53-5 \sqrt{3}
Option A (−53-5 \sqrt{3}) is correct because using the properties of the tangent function, the expression simplifies to −5tan⁡π3-5 \tan \frac{\pi}{3}, which equals −53-5\sqrt{3}.
563 Easy
563 of 581
Analyzing +4 | -1
The number of solutions of the equation tan⁡  x+sec⁡  x=2  cos⁡  x\tan \,\,x + \sec \,\,x = 2\,\,\cos \,\,x 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 the equation tan⁡x+sec⁡x=2cos⁡x\tan x + \sec x = 2 \cos x has two solutions in the interval [0,2π][0, 2\pi].
564 Medium
564 of 581
Application +4 | -1
The value of tan⁡2θsec⁡2θ(cot⁡2θ−cos⁡2θ)\tan ^{2} \theta \sec ^{2} \theta\left(\cot ^{2} \theta-\cos ^{2} \theta\right) is
View answer and explanation
Correct option: B
1
Option B is correct because simplifying the given expression using trigonometric identities leads to the value 1.
565 Easy
565 of 581
Application +4 | -1
If θ∈[0,2π]\theta \in[0,2 \pi] then number of ordererd pairs (x,θ)(x, \theta) which satisfies x4−2x2cos⁡θ+1=0x^4-2 x^2 \cos \theta+1=0 is/are
View answer and explanation
Correct option: D
4
Option D (4) is correct because the equation can be simplified to find the possible values of x and theta that satisfy it.
566 Medium
566 of 581
Analyzing +4 | -1
The equation cos⁡4x−(λ+2)cos⁡2x−(λ+3)=0{\cos ^4}x - \left( {\lambda + 2} \right){\cos ^2}x - \left( {\lambda + 3} \right) = 0 possesses a solution, if
View answer and explanation
Correct option: C
−3≤λ≤−2- 3 \le \lambda \le - 2
Option C (−3≤λ≤−2- 3 \le \lambda \le - 2) is correct because it is the range of λ\lambda that satisfies the condition 0≤cos⁡2x ≤10 \le {\cos ^2}x\, \le 1.
567 Medium
567 of 581
Application +4 | -1
The complete solution of the equation 7cos⁡2x+sin⁡xcos⁡x−3=07 \cos ^2 x+\sin x \cos x-3=0 is given by:
View answer and explanation
Correct option: D
nπ+3π4,kπ+tan⁡−1(43)(k,n∈Z){n} \pi+\frac{3 \pi}{4}, {k} \pi+\tan -1\left(\frac{4}{3}\right)({k}, {n} \in {Z})
Option D (nπ+3π4,kπ+tan⁡−1(43)(k,n∈Z){n} \pi+\frac{{3 \pi}}{{4}}, {k} \pi+\tan -1(\frac{{4}}{{3}})({k}, {n} \in {Z})) is correct because it represents the complete solution set obtained by solving the trigonometric equation after converting it to a quadratic in tan x.
568 Medium
568 of 581
Application +4 | -1
Solution of the inequality tan⁡(x+π3)≥1  \tan \left( {x + \frac{\pi }{3}} \right) \ge 1\; is
View answer and explanation
Correct option: B
(nπ−π12,nπ+π6)\left( {n\pi - \frac{\pi }{{12}},n\pi + \frac{\pi }{6}} \right)
Option B is correct because it represents the solution set for the given inequality, derived by considering the range of the tangent function.
569 Easy
569 of 581
Analyzing +4 | -1
The value(s) of x∈[−2π,2π]x \in[-2 \pi, 2 \pi] such that sin⁡x+icos⁡x1+i,i=−1\frac{\sin x+i \cos x}{1+i}, i=\sqrt{-1} is purely imaginary, is/are given by
View answer and explanation
Correct option: A
nπ−π4n \pi-\frac{\pi}{4}
Option A (nπ−π4n \pi - \frac{\pi}{4}) is correct because it satisfies the condition where the real part of the given expression is zero, making it purely imaginary.
570 Medium
570 of 581
Analyzing +4 | -1
The values of θ\theta, for which the system (sin⁡3θ)x−y+z=0,(cos⁡2θ)x+4y+3z=0(\sin 3 \theta) x-y+z=0,(\cos 2 \theta) x+4 y+3 z=0 and 2x+7y+7z=02 x+7 y+7 z=0, has a non-trivial solution, are
View answer and explanation
Correct option: A
nπ,nπ+(−1)nπ6n \pi, n \pi+(-1)^n \frac{\pi}{6}
Option A (nπ,nπ+(−1)nπ6n \pi, n \pi+(-1)^n \frac{\pi}{6}) is correct because these are the values of θ that make the determinant of the coefficient matrix zero, leading to non-trivial solutions.
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