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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Medium

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Showing 151-180 of 294 questions

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151 Medium
151 of 294
Application +4 | -1
The solution set of (5+4cos⁡  θ)  (2cos⁡  θ+1)=0\left( {5 + 4\cos \,\,\theta } \right)\,\,\left( {2\cos \,\,\theta + 1} \right) = 0 in the interval [0,   2π]\left[ {0,\;\,2\pi } \right] is

View answer and explanation
Correct option: C
{2 π3,   4 π3}\left\{ {\frac{{2\,\pi }}{3},\,\;\frac{{4\,\pi }}{3}} \right\}
Option C ({2,π3,;4,π3}\{\frac{{2\\,\pi }}{{3}},\\;\frac{{4\\,\pi }}{{3}}\}) is correct because these are the only solutions in the given interval that satisfy the equation.
152 Medium
152 of 294
Application +4 | -1
If cot⁡(α+β)=0,  then  sin⁡(α+2β)\cot \left( {\mathrm{\alpha } + \mathrm{\beta }} \right) = 0,\mathrm{\;then}\,\,\sin \left( {\mathrm{\alpha } + 2\mathrm{\beta }} \right) is equal to

View answer and explanation
Correct option: A
sin⁡α\sin \mathrm{\alpha }
Option A (sin⁡α\sin \mathrm{\alpha }) is correct because, given the condition, the expression simplifies to sin⁡α\sin{\alpha} using trigonometric identities.
153 Medium
153 of 294
Application +4 | -1
If sin⁡A+cos⁡A=2\sin A+\cos A=\sqrt{2}, then the value of cos⁡2A\cos ^2 A is
View answer and explanation
Correct option: B
1/21 / 2
Option B (1/2) is correct because by solving the given equation for A, we find A = π/4\pi/4, and then cos⁡2A\cos^2 A can be easily calculated.
154 Medium
154 of 294
Analyzing +4 | -1
If αcos⁡23θ+βcos⁡4θ=16cos⁡6θ+9cos⁡2θ\alpha \cos ^2 3 \theta+\beta \cos ^4 \theta=16 \cos ^6 \theta+9 \cos ^2 \theta is an identity, then
View answer and explanation
Correct option: B
α=1,β=24\alpha=1, \beta=24
Option B (α=1,β=24\alpha=1, \beta=24) is correct because substituting these values into the given identity and simplifying leads to a valid equation.
155 Medium
155 of 294
Analyzing +4 | -1
If 0≤a≤3,0≤b≤30 \leq a \leq 3,0 \leq b \leq 3 and the equation x2+4+3cos⁡(ax+b)=2xx^2+4+3 \cos (a x+b)=2 x has atleast one solution, then the value of a+ba+b is
View answer and explanation
Correct option: C
π\pi
Option C (π\pi) is correct because the equation has at least one solution only if cos⁡(ax+b)=−1\cos(ax+b) = -1 when x=1x=1, leading to a+b=πa+b = \pi.
156 Medium
156 of 294
Application +4 | -1
If cos⁡(θ−α)=a\cos (\theta-\alpha)=a and sin⁡(θ−β)=b\sin (\theta-\beta)=b, then cos⁡2(α−β)+2absin⁡(α−β)\cos ^2(\alpha-\beta)+2 a b \sin (\alpha-\beta) is equal to
View answer and explanation
Correct option: C
a2+b2a^2+b^2
The correct answer is a2+b2a^2+b^2 because the expression simplifies to a2+b2a^2 + b^2 using trigonometric identities and the given substitutions.
157 Medium
157 of 294
Application +4 | -1
If x=cos⁡15∘x=\cos 15^{\circ}, then which of the following is correct ?
View answer and explanation
Correct option: D
16x4−16x2+1=016 x^4-16 x^2+1=0
Option D (16x4−16x2+1=016 x^4-16 x^2+1=0) is correct because it is derived from the given condition x=cos⁡15∘x = \cos 15^{\circ} by algebraic manipulation.
158 Medium
158 of 294
Analyzing +4 | -1
In a △ABC,tan⁡A2\triangle A B C, \tan \frac{A}{2} and tan⁡B2\tan \frac{B}{2} satisfy 6x2−5x+1=06 x^2-5 x+1=0. Then
View answer and explanation
Correct option: C
a2+b2=c2a^2+b^2=c^2
Option C (a2+b2=c2a^2+b^2=c^2) is correct because the given condition implies that the triangle is right-angled at C, hence satisfying the Pythagorean theorem.
159 Medium
159 of 294
Analyzing +4 | -1
If 2  sin⁡  A2=1+sin⁡  A+1−sin⁡  A2\,\,\sin \,\,\frac{A}{2} = \sqrt {1 + \sin \,\,A} + \sqrt {1 - \sin \,\,A}, then A2\frac{A}{2} lies between
View answer and explanation
Correct option: A
2n  π+π42n\;\pi + \frac{\pi }{4} and 2n  π+3 π4,   n∈Z2n\;\pi + \frac{{3\,\pi }}{4},\;\,n \in Z
Option A (2n;π+π42n\\;\pi + \frac{\pi }{4} and 2n;π+3,π4,   n∈Z2n\\;\pi + \frac{{3\\,\pi }}{4},\;\,n \in Z) is correct because it defines the interval where A2\frac{A}{2} lies, based on the given equation and trigonometric properties.
160 Medium
160 of 294
Application +4 | -1
If D=∣1cos⁡θ1−sin⁡θ1−cos⁡θ−1sin⁡θ1∣D=\left|\begin{array}{ccc}1 & \cos \theta & 1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1\end{array}\right|

f(x)f(x) 52θ5^2 \theta then DD lies in the interval
View answer and explanation
Correct option: C
[2−2,2+2][2-\sqrt{2}, 2+\sqrt{2}]
Option C ([2−2,2+2][2-\sqrt{2}, 2+\sqrt{2}]) is correct because by expanding the determinant and simplifying the expression, we can find the range of values for D.
161 Medium
161 of 294
Analyzing +4 | -1
If A,BA, B and CC are angles of anyone of the triangle such that tan⁡A+tan⁡B+tan⁡C=100\tan A+\tan B+\tan C=100. Then,
View answer and explanation
Correct option: A
There exist exactly three non-similar isosceles triangles
Option A (There exist exactly three non-similar isosceles triangles) is correct because the equation x3−50x2+50=0x^3-50 x^2+50=0 has three distinct real roots, implying three different values for angle A.
162 Medium
162 of 294
Analyzing +4 | -1
If x=sin⁡(α−β)sin⁡(γ−δ),  y=sin⁡(β−γ)sin⁡(α−δ)x = \sin \left( {\alpha - \beta } \right)\sin \left( {\gamma - \delta } \right),\,\,y = \sin \left( {\beta - \gamma } \right)\sin \left( {\alpha - \delta } \right) and z=sin⁡(γ−α)sin⁡(β−δ),z = \sin \left( {\gamma - \alpha } \right)\sin \left( {\beta - \delta } \right), then
View answer and explanation
Correct option: A
x+y+z=0x+y+z=0
Option A (x+y+z=0x+y+z=0) is correct because when the expressions for x, y, and z are expanded and added, the terms cancel out, resulting in zero.
163 Medium
163 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: B
No real roots
The equation has no real roots because the value of esin⁡xe^{\sin x} cannot be negative or less than 1, and ln⁡(2+5)>1\ln(2+\sqrt{5}) > 1, which is outside the range of sin⁡x\sin x.
164 Medium
164 of 294
Application +4 | -1
The solution of tan⁡  2θ  tan⁡  θ=1\tan \,\,2\theta \,\,\tan \,\,\theta = 1 is 22\mathrm{22}
View answer and explanation
Correct option: B
(6n  ±  1)π6\left( {6n\,\, \pm \,\,1} \right)\frac{\pi }{6}
Option B is the correct solution because it represents the general solution to the trigonometric equation tan⁡2θtan⁡θ=1\tan 2\theta \tan \theta = 1 after simplification and solving for θ\theta.
165 Medium
165 of 294
Analyzing +4 | -1
If sin⁡α=15\sin \alpha=\frac{1}{\sqrt{5}} and sin⁡β=35\sin \beta=\frac{3}{5}, then β−α\beta-\alpha lies in the interval
View answer and explanation
Correct option: A
(0,π4)∪(3π4,π)\left(0, \frac{\pi}{4}\right) \cup\left(\frac{3 \pi}{4}, \pi\right)
Option A ((0,π4)∪(3π4,π)\left(0, \frac{{\pi}}{{4}}\right) \cup\left(\frac{{3 \pi}}{{4}}, \pi\right)) is correct because the value of sin⁡(β−α)\sin(\beta - \alpha) is positive and less than 12\frac{{1}}{{\sqrt{{2}}}}, which restricts the interval for β−α\beta - \alpha.

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166 Medium
166 of 294
Analyzing +4 | -1
The equation 3sin⁡x+cos⁡x=4\sqrt{3} \sin x+\cos x=4 has
View answer and explanation
Correct option: D
No solution
Option D (No solution) is correct because the maximum value of 3sin⁡x+cos⁡x\sqrt{3} \sin x + \cos x is 2, which is less than 4.
167 Medium
167 of 294
Analyzing +4 | -1
If −π2≤x≤π2- \frac{\pi }{2} \le x \le \frac{\pi }{2}, then the two curves y=cos⁡xy = \cos x and y=sin⁡3xy = \sin 3x intersect at
View answer and explanation
Correct option: A
(π4, 12)   and   (π8,  cos⁡  π8)\left( {\frac{\pi }{4},\,\frac{1}{{\sqrt 2 }}} \right)\;\,\mathrm{and\;}\,\left( {\frac{\pi }{8},\,\,\cos \,\,\frac{\pi }{8}} \right)
Option A is correct because the intersection points are found by equating the two functions and solving for x, then finding the corresponding y values.
168 Medium
168 of 294
Application +4 | -1
If α\alpha is a root of 25cos⁡2θ+5cos⁡θ−12=025 \cos ^2 \theta+5 \cos \theta-12=0, π2<α<π\frac{\pi}{2} < \alpha < \pi, then sin⁡2α\sin 2 \alpha is equal to
View answer and explanation
Correct option: B
−2425-\frac{24}{25}
Option B (−2425-\frac{24}{25}) is correct because solving the quadratic equation for cos⁡α\cos \alpha and using the given range for α\alpha allows us to find sin⁡α\sin \alpha and then sin⁡2α\sin 2\alpha.
169 Medium
169 of 294
Analyzing +4 | -1
If 0≤x≤3π,0≤y≤3π0 \leq x \leq 3 \pi, 0 \leq y \leq 3 \pi and cos⁡xsin⁡y=1\cos x \sin y=1, then the possible number of values of the ordered pair (x,y)(x, y) is
View answer and explanation
Correct option: A
6
Option B (3) is correct because there are three ordered pairs (x, y) that satisfy the given equation within the specified intervals.
170 Medium
170 of 294
Analyzing +4 | -1
If α,β,γ\alpha, \beta, \gamma and δ\delta are the solutions of the equation tan⁡(θ+π4)=3tan⁡3θ\tan \left(\theta+\frac{\pi}{4}\right)=3 \tan 3 \theta, no two of which have equal tangents, then the value of tan⁡α+tan⁡β+tan⁡γ+tan⁡δ\tan \alpha+\tan \beta+\tan \gamma+\tan \delta is
View answer and explanation
Correct option: D
0
Option D (0) is correct because the sum of the roots of the quartic equation in tan⁡θ\tan \theta is 0.
171 Medium
171 of 294
Application +4 | -1
If A+B=π4A+B=\frac{\pi}{4}, where A,B∈R+A, B \in R^{+}, then the minimum value of (1+tan⁡A)(1+tan⁡B)(1+\tan A)(1+\tan B) is
View answer and explanation
Correct option: A
2
Option A (2) is correct because the minimum value of the expression is found by using the given condition and trigonometric identities to simplify it to a constant value of 2.
172 Medium
172 of 294
Application +4 | -1
If 3  cos⁡ x≠2  sin⁡x3\,\,\cos \,x \ne 2\,\,\sin x, then the general solution of sin⁡2x−cos⁡  2x=2−sin⁡  2x{\sin ^2}x - \cos \,\,2x = 2 - \sin \,\,2x is
View answer and explanation
Correct option: C
(4n±1)π2,   n∈Z\left( {4n \pm 1} \right)\frac{\pi }{2},\,\;n \in Z
Option C ((4n±1)π2,; n∈Z\left( {4n \pm 1} \right)\frac{\pi }{{2}},\\;\,n \in Z) is correct because solving the trigonometric equation leads to this general solution.
173 Medium
173 of 294
Application +4 | -1
If cos⁡θ=12(x+1x)\cos \theta=\frac{1}{2}\left(x+\frac{1}{x}\right), then 12(x2+1x2)\frac{1}{2}\left(x^2+\frac{1}{x^2}\right) is equal to
View answer and explanation
Correct option: B
cos⁡2θ\cos 2 \theta
Option B (cos⁡2θ\cos 2 \theta) is correct because it is derived from the given equation by squaring and using the double angle formula for cosine.
174 Medium
174 of 294
Analyzing +4 | -1
Let α,  β\mathrm{\alpha },\,\,\mathrm{\beta } be any two positive values of xx for which 2cos⁡  x,  ∣cos⁡  x∣     and   1−3  cos⁡2x2\cos \,\,x,\;|\cos \,\,x|\;\;\,\mathrm{and}\;\,1 - 3\,\,{\cos ^2}x are in GP. The minimum value of ∣α−β∣\left| {\mathrm{\alpha } - \mathrm{\beta }} \right| is
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the minimum value of |α−β\alpha - \beta| is π6\frac{\pi}{6}, which is not among the given options.
175 Medium
175 of 294
Application +4 | -1
If sin⁡x+sin⁡2x=1\sin x+\sin ^2 x=1, then the value of cos⁡12x+3cos⁡10x+3cos⁡8x+cos⁡6x\cos ^{12} x+3 \cos ^{10} x+3 \cos ^8 x+\cos ^6 x is
View answer and explanation
Correct option: A
1
Option A (1) is correct because the given expression simplifies to 1 using the trigonometric identity sin⁡x=cos⁡2x\sin x = \cos^2 x.
176 Medium
176 of 294
Understanding +4 | -1
The equation asin⁡x+bcos⁡x=ca \sin x+b \cos x=c, where ∣c∣>a2+b2|c|>\sqrt{a^2+b^2} has
View answer and explanation
Correct option: C
No solution
The correct answer is No solution because the given condition ∣c∣>a2+b2|c| > \sqrt{{a^2+b^2}} implies that the value of c is outside the range of asin⁡x+bcos⁡xa \sin x + b \cos x.
177 Medium
177 of 294
Application +4 | -1
If tan⁡ (πcos⁡  θ)=cot⁡ (π  sin⁡  θ)\tan \,(\mathrm{\pi }\cos \,\,\mathrm{\theta }) = \cot \,(\mathrm{\pi }\,\,\sin \,\,\mathrm{\theta }), then the value of cos⁡  (θ−π4)\cos \,\,\left( {\mathrm{\theta } - \frac{\mathrm{\pi }}{4}} \right) is equal to
View answer and explanation
Correct option: A
12 2\frac{1}{{2\,\sqrt 2 }}
The correct answer is A (122\frac{1}{{2\sqrt 2 }}) because solving the given equation leads to sin⁡θ+cos⁡θ=12\sin \theta + \cos \theta = \frac{1}{2}, which simplifies to the desired expression.
178 Medium
178 of 294
Analyzing +4 | -1
The number of points of intersection of the curves y=cos⁡x,y=sin⁡3xy=\cos x, y=\sin 3 x, if −π2≤x≤π2-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, is
View answer and explanation
Correct option: A
3
Option A, 3, is correct because there are three points of intersection between the curves y=cos⁡xy=\cos x and y=sin⁡3xy=\sin 3x within the given interval.
179 Medium
179 of 294
Analyzing +4 | -1
For 0<ϕ<π20 < \phi < \frac{\pi}{2}, if x=∑n=0∞cos⁡2nϕ,y=∑n=0∞sin⁡2nϕx=\sum\limits_{{n=0}}^{\infty} \cos ^{2 n} \phi, y=\sum\limits_{{n=0}}^{\infty} \sin ^{2 n} \phi and z=∑n=0∞cos⁡2nϕsin⁡2nϕz=\sum\limits_{{n=0}}^{\infty} \cos ^{2 n} \phi \sin ^{2 n} \phi, then
View answer and explanation
Correct option: C
xyz=x+y+zx y z=x+y+z
Option D (xyz=yz+xx y z=y z+x) is incorrect. The correct option is C (xyz=x+y+zx y z=x+y+z) because of the relationship between the sums and the trigonometric functions.
180 Medium
180 of 294
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.
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