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JEE • MATHEMATICS • TRIGONOMETRY RATIOS

Trigonometry ratios practice · Mixed difficulty

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Showing 151-180 of 394 questions

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151 Easy
151 of 394
Application +4 | -1
If tan⁡x=ba\tan x=\frac{b}{a} then a+ba−b+a−ba+b\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}} is equal to
View answer and explanation
Correct option: B
2cos⁡xcos⁡2x\frac{2 \cos x}{\sqrt{\cos 2 x}}
Option B (2cos⁡xcos⁡2x\frac{{2 \cos x}}{{\sqrt{{\cos 2 x}}}}) is correct because the expression simplifies to this form after algebraic manipulation and using trigonometric identities.
152 Easy
152 of 394
Understanding +4 | -1
Consider the statements given below:

I. 2cos⁡x⋅cos⁡y=cos⁡(x+y)−cos⁡(x−y)2 \cos x \cdot \cos y=\cos (x+y)-\cos (x-y).

II. −2sin⁡x⋅sin⁡y=cos⁡(x+y)−cos⁡(x−y)-2 \sin x \cdot \sin y=\cos (x+y)-\cos (x-y).

III. 2sin⁡x⋅cos⁡y=sin⁡(x+y)−sin⁡(x−y)2 \sin x \cdot \cos y=\sin (x+y)-\sin (x-y).

IV. 2cos⁡x⋅sin⁡y=sin⁡(x+y)+sin⁡(x−y)2 \cos x \cdot \sin y=\sin (x+y)+\sin (x-y).

Choose the correct statements.
View answer and explanation
Correct option: B
II is correct
Option B is correct because the identity −2sin⁡x⋅sin⁡y=cos⁡(x+y)−cos⁡(x−y)-2 \sin x \cdot \sin y = \cos(x+y) - \cos(x-y) is a standard trigonometric identity.
153 Medium
153 of 394
Application +4 | -1
The value of (1+tan⁡11∘)(1+tan⁡34∘)(1+tan⁡17∘)(1+tan⁡28∘)\frac{\left(1+\tan 11^{\circ}\right)\left(1+\tan 34^{\circ}\right)}{\left(1+\tan 17^{\circ}\right)\left(1+\tan 28^{\circ}\right)} is equal to
View answer and explanation
Correct option: A
1
Option A (1) is correct because the expression simplifies to 1 using the trigonometric identity (1+tanA)(1+tanB)=2 when A+B=45 degrees.
154 Medium
154 of 394
Application +4 | -1
1−sin⁡A1+sin⁡A=……\sqrt{\frac{1-\sin \mathrm{A}}{1+\sin \mathrm{A}}}=\ldots \ldots    −π2<A<π2\,\,\,-\frac{\pi}{2} < \mathrm{A} < \frac{\pi}{2}
View answer and explanation
Correct option: B
sec⁡A−tan⁡A\sec \mathrm{A}-\tan \mathrm{A}
Option B (sec⁡A−tan⁡A\sec A - \tan A) is correct because simplifying the given expression using trigonometric identities leads to this result, considering the given range of A.
155 Hard
155 of 394
Application +4 | -1
If P=12  sin⁡2θ  +  13  cos⁡2θP = \frac{1}{2}\,\,{\sin ^2}\theta \,\, + \,\,\frac{1}{3}\,\,{\cos ^2}\theta, then
View answer and explanation
Correct option: A
13≤P≤12\frac{1}{3} \le P \le \frac{1}{2}
Option A (13≤P≤12\frac{1}{3} \le P \le \frac{1}{2}) is correct because by rewriting P in terms of cos²θ and using the range of cos²θ, we can find the range of P.
156 Medium
156 of 394
Application +4 | -1
If sin⁡(A+B+C)=1,tan⁡(A−B)=13\sin (A+B+C)=1, \tan (A-B)=\frac{1}{\sqrt{3}} and sec⁡(A+C)=2\sec (A+C)=2, then
View answer and explanation
Correct option: C
A=60∘,B=30∘,C=0∘A=60^{\circ}, B=30^{\circ}, C=0^{\circ}
Option C (A=60∘,B=30∘,C=0∘A=60^{\circ}, B=30^{\circ}, C=0^{\circ}) is correct because solving the system of equations derived from the given trigonometric relations yields these values for A, B, and C.
157 Easy
157 of 394
Application +4 | -1
The least value of cosec⁡2x+25sec⁡2x\operatorname{cosec}^2 x+25 \sec ^2 x is
View answer and explanation
Correct option: D
36
Option D (36) is correct because it is the minimum value obtained by completing the square and using the AM-GM inequality.
158 Medium
158 of 394
Understanding +4 | -1
cos⁡  1∘+cos⁡  2∘+cos⁡  3∘+...+cos⁡  180∘\cos \,\,1^\circ + \cos \,\,2^\circ + \cos \,\,3^\circ + ... + \cos \,\,180^\circ is equal to
View answer and explanation
Correct option: D
−1- 1
Option D (-1) is correct because the sum telescopes due to the symmetry of cosine around 90 degrees, with cos(180) being -1.
159 Medium
159 of 394
Application +4 | -1
If A=130∘A = 130^\circ and x=sin⁡A+cos⁡Ax = \sin A + \cos A, then
View answer and explanation
Correct option: A
x>0x > 0
Option A (x>0x > 0) is correct because sin⁡130∘+cos⁡130∘\sin 130^\circ + \cos 130^\circ simplifies to sin⁡50∘−sin⁡40∘\sin 50^\circ - \sin 40^\circ, which is positive since sin⁡50∘>sin⁡40∘\sin 50^\circ > \sin 40^\circ.
160 Medium
160 of 394
Application +4 | -1
If sin⁡A−6cos⁡A=7cos⁡A\sin A-\sqrt{6} \cos A=\sqrt{7} \cos A, then cos⁡A+6sin⁡A\cos A+\sqrt{6} \sin A is equal to
View answer and explanation
Correct option: B
7sin⁡A\sqrt{ } 7 \sin A
Option B (7sin⁡A\sqrt{7} \sin A) is correct because by manipulating the given equation, we can express cos⁡A\cos A in terms of sin⁡A\sin A and then substitute to find the value of cos⁡A+6sin⁡A\cos A + \sqrt{6} \sin A.
161 Easy
161 of 394
Application +4 | -1
The greatest and least value of sin⁡  x  cos⁡  x\sin \,\,x\,\,\cos \,\,x are respectively
View answer and explanation
Correct option: B
12,  −12\frac{1}{2},\,\, - \frac{1}{2}
Option B (12,  −12\frac{1}{2},\,\, - \frac{1}{2}) is correct because it represents the maximum and minimum values of the function 12sin⁡2x\frac{1}{2}\sin 2x.
162 Medium
162 of 394
Application +4 | -1
The minimum value of sin⁡θcos⁡θ\sin \theta \cos \theta is
View answer and explanation
Correct option: C
−12-\frac{1}{2}
Option C (-12\frac{{1}}{{2}}) is correct because the minimum value of sin⁡θcos⁡θ\sin \theta \cos \theta is -12\frac{{1}}{{2}}, which can be found using trigonometric identities or calculus.
163 Medium
163 of 394
Application +4 | -1
The value of sin⁡47∘+sin⁡61∘−sin⁡11∘−sin⁡25∘=\sin 47^{\circ}+\sin 61^{\circ}-\sin 11^{\circ}-\sin 25^{\circ}=
View answer and explanation
Correct option: D
cos⁡7∘\cos 7^{\circ}
Option D (cos⁡7∘\cos 7^{\circ}) is the correct answer because after applying trigonometric identities, the expression simplifies to cos⁡7∘\cos 7^{\circ}.
164 Easy
164 of 394
Application +4 | -1
If A=130∘A = 130^\circ and x=sin⁡A+cos⁡Ax = \sin A + \cos A, then
View answer and explanation
Correct option: A
x>0x > 0
Option A (x>0x > 0) is correct because sin⁡130∘\sin 130^\circ is positive and greater than the absolute value of cos⁡130∘\cos 130^\circ, which is negative.
165 Easy
165 of 394
Application +4 | -1
If the value of 3cos⁡36∘+5sin⁡18∘5cos⁡36∘−3sin⁡18∘\frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}} is a5−bc,\frac{\mathrm{a} \sqrt{5}-\mathrm{b}}{\mathrm{c}}, where a,b,c\mathrm{a}, \mathrm{b}, \mathrm{c} are natural numbers and gcd⁡(a,c)=1,\operatorname{gcd}(\mathrm{a}, \mathrm{c})=1, then a+b+c\mathrm{a}+\mathrm{b}+\mathrm{c} is equal to:
View answer and explanation
Correct option: C
52
Option C (52) is correct because after simplifying the given expression using trigonometric values and rationalizing the denominator, the values of a, b, and c can be determined, and their sum is 52.

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166 Medium
166 of 394
Analyzing +4 | -1
The range of cos⁡θ(sin⁡θ+sin⁡2θ+sin⁡2α)\cos \theta\left(\sin \theta+\sqrt{\sin ^2 \theta+\sin ^2 \alpha}\right) is
View answer and explanation
Correct option: A
[−1+sin⁡2α,1+sin⁡2α]\left[-\sqrt{1+\sin ^2 \alpha}, \sqrt{1+\sin ^2 \alpha}\right]
Option A is correct because the range is found by solving the trigonometric equation and considering the discriminant of the resulting quadratic equation in terms of tan θ.
167 Easy
167 of 394
Analyzing +4 | -1
If 0<x<π2,  0 < x < \frac{\pi }{2},\; then minimum value of cos3xsin⁡x+sin⁡3xcos⁡x  \frac{{\mathrm{co}{\mathrm{s}^3}x}}{{\sin x}} + \frac{{{{\sin }^3}x}}{{\cos x}}\; is
View answer and explanation
Correct option: D
1
The correct answer is D. Using AM-GM inequality, we can find that the minimum value of the expression is 1.
168 Easy
168 of 394
Application +4 | -1
If   sin⁡  Asin⁡  B=  32  and    cos⁡  Acos⁡  B=52,  0<A,  B<π2  \;\frac{{\sin \,\,A}}{{\sin \,\,B}} = \;\frac{{\sqrt 3 }}{2}\,\,\mathrm{and}\;\,\,\frac{{\cos \,\,A}}{{\cos \,\,B}} = \frac{{\sqrt 5 }}{2},\,\,0 < A,\,\,B < \frac{\pi }{2}\;, then
View answer and explanation
Correct option: A
tan⁡A=35\tan A = \frac{{\sqrt 3 }}{{\sqrt 5 }}
Option A (tan⁡A=35\tan A = \frac{{\sqrt 3 }}{{\sqrt 5 }}) is correct because, by dividing the given equations, we can directly find the value of tan⁡A\tan A.
169 Easy
169 of 394
Application +4 | -1
In a △ABC,∠A=π2\triangle A B C, \angle A=\frac{\pi}{2}, then cos⁡2B+cos⁡2C\cos ^2 B+\cos ^2 C is equal to
View answer and explanation
Correct option: B
1
Option B (1) is correct because when ∠A=π2\angle A = \frac{\pi}{2}, angles B and C are complementary, so cos⁡C=sin⁡B\cos C = \sin B, and cos⁡2B+sin⁡2B=1\cos^2 B + \sin^2 B = 1.
170 Easy
170 of 394
Application +4 | -1
If 2cos⁡θ+sin⁡θ=1,(θ≠(4k+1)π/22 \cos \theta+\sin \theta=1,(\theta \neq(4 k+1) \pi / 2, k∈k \in I) then 7cos⁡θ+6sin⁡θ7 \cos \theta+6 \sin \theta is equal to:
View answer and explanation
Correct option: B
2
The correct answer is B (2) because by solving the given equation for sin θ and cos θ, we can substitute these values into the expression 7cos θ + 6sin θ to obtain the result.
171 Easy
171 of 394
Application +4 | -1
The value of sin⁡10∘+sin⁡20∘+sin⁡30∘+…+sin⁡360∘\sin 10^\circ + \sin 20^\circ + \sin 30^\circ + \ldots + \sin 360^\circ is
View answer and explanation
Correct option: B
0
Option B (0) is correct because the sines of angles equally spaced around the unit circle sum to zero due to symmetry.
172 Easy
172 of 394
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−1\cos ^{12} x+3 \cos ^{10} x+3 \cos ^8 x+\cos ^6 x-1 is equal to
View answer and explanation
Correct option: A
0
Option A (0) is correct because by substituting sin⁡x=cos⁡2x\sin x = \cos^2 x into the given expression, it simplifies to 0.
173 Easy
173 of 394
Understanding +4 | -1
The value of   cos⁡  1∘  cos⁡  2∘  cos⁡  3∘…cos⁡  100∘  \mathrm{\;}\cos \,\,1^\circ \,\,\cos \,\,2^\circ \,\,\cos \,\,3^\circ \ldots \cos \,\,100^\circ \mathrm{\;} is equal to
View answer and explanation
Correct option: C
0\mathrm{0}
Option C (0) is correct because the product includes cos(90°), which equals 0, thus making the entire product equal to 0.
174 Medium
174 of 394
Application +4 | -1
If sin⁡2θ=x2+y2+12x\sin ^2 \theta=\frac{x^2+y^2+1}{2 x}, then xx must be
View answer and explanation
Correct option: C
1
Option C (1) is correct because the given equation implies that xx must be 1 to satisfy the condition 0≤sin⁡2θ≤10 \leq \sin^2 \theta \leq 1.
175 Medium
175 of 394
Understanding +4 | -1
How many times do the hands of the watch form an angle of 180 degree during a complete day?
View answer and explanation
Correct option: B
22 times
Option B (22 times) is correct because the hands of a watch form a 180-degree angle 22 times in a 24-hour period.
176 Easy
176 of 394
Application +4 | -1
sin⁡75∘=\sin 75^{\circ}=
View answer and explanation
Correct option: B
3+122\frac{\sqrt{3}+1}{2 \sqrt{2}}
Option B (3+122\frac{{\sqrt{{3}}+1}}{{2 \sqrt{{2}}}}) is correct because it is derived from using the sine addition formula sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B) = \sin A \cos B + \cos A \sin B with A = 45 and B = 30.
177 Easy
177 of 394
Application +4 | -1
If x+y=zx+y=z, then cos⁡2x+cos⁡2y+cos⁡2z−2cos⁡xcos⁡ycos⁡z\cos ^2 x+\cos ^2 y+\cos ^2 z-2 \cos x \cos y \cos z is equal to
View answer and explanation
Correct option: D
1
Option D (3+55\frac{{\sqrt{{3}} + \sqrt{{5}}}}{{\sqrt{{5}}}}) is correct because it is derived by using the given ratios to find tan⁡A\tan A and tan⁡B\tan B, then summing them.
178 Easy
178 of 394
Understanding +4 | -1
Which one of the following is possible?
View answer and explanation
Correct option: C
tan⁡  θ=45\tan \,\,\theta = 45
Option C (tan⁡ θ=45\tan \,\theta = 45) is possible because the range of the tangent function is all real numbers.
179 Medium
179 of 394
Application +4 | -1
The value of cos⁡480∘sin⁡150∘+sin⁡60∘⋅cos⁡390∘\cos 480^{\circ} \sin 150^{\circ}+\sin 60^{\circ} \cdot \cos 390^{\circ} is equal to
View answer and explanation
Correct option: C
1/21 / 2
Option C (1/21 / 2) is correct because substituting the values of the trigonometric functions and simplifying the expression leads to the result 1/21/2.
180 Medium
180 of 394
Analyzing +4 | -1
sec⁡  θ=a2+  b2a2−  b2\sec \,\,\theta = \frac{{{a^2} + \,\,{b^2}}}{{{a^2} - \,\,{b^2}}}, where a,   b, ∈Ra,\,\;b,\, \in R gives real values of θ\theta if and only if

View answer and explanation
Correct option: B
∣ a ∣≠∣ b ∣≠0\left| {\,a\,} \right| \ne \left| {\,b\,} \right| \ne 0
Option B (\left| {\,a\} \right| \ne \left| {\,b\} \right| \ne 0\left| {\,a\} \right| \ne \left| {\,b\} \right| \ne 0) is correct because it ensures that the value of sec⁡θ\sec \theta is real and either greater than or equal to 1, or less than or equal to -1.
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