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JEE • MATHEMATICS • TRIGONOMETRY RATIOS

Trigonometry ratios practice · Mixed difficulty

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Showing 301-330 of 394 questions

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301 Medium
301 of 394
Application +4 | -1
In a △ABC,cos⁡A=35\triangle A B C, \cos A=\frac{3}{5} and cos⁡B=513\cos B=\frac{5}{13}. The value of cos⁡C\cos C can be
View answer and explanation
Correct option: C
3365\frac{33}{65}
Option C (3365\frac{33}{65}) is correct because it is the value of cos C that satisfies the trigonometric relationships within a triangle, given the values of cos A and cos B.
302 Medium
302 of 394
Application +4 | -1
If 3π4<a<π\frac{3 \pi}{4} < a < \pi, then cosec⁡2α+2cot⁡α=\sqrt{\operatorname{cosec}^2 \alpha+2 \cot \alpha}=
View answer and explanation
Correct option: C
−1−cot⁡α-1-\cot \alpha
Option C (−1−cot⁡α-1-\cot \alpha) is correct because within the given range for α\alpha, 1+cot⁡α1 + \cot \alpha is negative, so the square root evaluates to the negative of that expression.
303 Easy
303 of 394
Application +4 | -1
If tan⁡A=1−cos⁡Bsin⁡B\tan \mathrm{A}=\frac{1-\cos \mathrm{B}}{\sin \mathrm{B}} then tan⁡2 A=\tan 2 \mathrm{~A}=
View answer and explanation
Correct option: A
tan⁡B\tan B
Option A (tan⁡B\tan B) is correct because by using the half-angle formula for tangent, we can simplify the given expression and find that tan⁡A=tan⁡B2\tan A = \tan \frac{B}{2}, which leads to tan⁡2A=tan⁡B\tan 2A = \tan B.
304 Medium
304 of 394
Analyzing +4 | -1
The greatest value of sin⁡3x+cos⁡3x{\sin ^3}x + {\cos ^3}x is
View answer and explanation
Correct option: A
1
Option A (1) is correct because the maximum value of the function is 1, which can be found by analyzing its derivative and critical points.
305 Easy
305 of 394
Application +4 | -1
If    θ\mathrm{If}\,\,\;\mathrm{\theta } is an acute angle and tan⁡  θ=17\tan \,\,\mathrm{\theta } = \frac{1}{{\sqrt 7 }}, then the value of cosec2 θ−sec⁡2  θcosec2 θ+sec⁡2  θ\frac{{\mathrm{cose}{\mathrm{c}^2}\,\mathrm{\theta } - {{\sec }^2}\,\,\mathrm{\theta }}}{{\mathrm{cose}{\mathrm{c}^2}\,\mathrm{\theta } + {{\sec }^2}\,\,\mathrm{\theta }}} is
View answer and explanation
Correct option: A
3/43\mathrm{ }/4
Option A (3/43\mathrm{ }/4) is correct because it is obtained by substituting the given value of tan⁡θ\tan \theta and using trigonometric identities to simplify the expression.
306 Easy
306 of 394
Understanding +4 | -1
Assertion: The second hand rotates through an angle of 180∘180^{\circ} in a minute.

Reason: The unit of measurement is degree in sexagesimal system.
View answer and explanation
Correct option: D
Assertion is incorrect, reason is correct
Assertion is incorrect, reason is correct because the second hand rotates 360 degrees in a minute, and the sexagesimal system uses degrees.
307 Hard
307 of 394
Analyzing +4 | -1
Let f(x)=cos⁡xsin⁡2xf\left( x \right) = \cos x\sin 2x Then, min⁡∣f(x):−π≤x≤π∣\min \left| {f\left( x \right): - \pi \le x \le \pi } \right| is greater than
View answer and explanation
Correct option: A
−97\frac{{ - 9}}{7}
Option A (−97\frac{{ - 9}}{7}) is correct because the minimum value of the function is greater than -9/7.
308 Easy
308 of 394
Application +4 | -1
If sin⁡  θ+cos⁡  θ=2cos⁡  θ\sin \,\,\theta + \cos \,\,\theta = \sqrt 2 \cos \,\,\theta then cos⁡  θ−sin⁡  θ\cos \,\,\theta - \sin \,\,\theta is equal to
View answer and explanation
Correct option: B
2  sin⁡θ\sqrt 2 \,\,\sin \theta
Option B (2 sin⁡θ\sqrt 2 \,\sin \theta) is correct because it is derived by squaring the given equation and using trigonometric identities to find the value of cos⁡θ−sin⁡θ\cos \theta - \sin \theta.
309 Easy
309 of 394
Application +4 | -1
The value of 1+sin⁡θ−cos⁡θ1+sin⁡θ+cos⁡θ\frac{1+\sin \theta-\cos \theta}{1+\sin \theta+\cos \theta} is
View answer and explanation
Correct option: B
tan⁡θ/2\tan \theta / 2
Option B (tan⁡θ/2\tan \theta / 2) is correct because simplifying the given expression using half-angle formulas results in tan⁡θ/2\tan \theta / 2.
310 Medium
310 of 394
Application +4 | -1
Sides of a △ABC\triangle A B C are in AP, if a<a< minimum {b,c}\{b, c\}, then cos⁡A\cos A is equal to
View answer and explanation
Correct option: D
4c−3b2c\frac{4 c-3 b}{2 c}
Option D (4c−3b2c\frac{4 c-3 b}{2 c}) is correct because by considering the possible arithmetic progressions and applying the cosine rule, this expression for cos⁡A\cos A is obtained.
311 Medium
311 of 394
Application +4 | -1
If acos⁡3α+3acos⁡αsin⁡2α=ma \cos ^3 \alpha+3 a \cos \alpha \sin ^2 \alpha=m and asin⁡3α+3acos⁡2αsin⁡α=na \sin ^3 \alpha+3 a \cos ^2 \alpha \sin \alpha=n, then (m+n)2/3+(m+n)^{2 / 3}+ (m−n)2/3(m-n)^{2 / 3} is equal to
View answer and explanation
Correct option: C
2a2/32 a^{2 / 3}
Option C (2a2/3a^{2/3}) is correct because, given the equations for m and n, we can find (m+n)(2/3) + (m-n)(2/3) by using the binomial expansion and trigonometric identities, which simplifies to 2a2/3a^{2/3}.
312 Medium
312 of 394
Application +4 | -1
If tan⁡θ+sin⁡θ=m\tan \theta+\sin \theta=m and tan⁡θ−sin⁡θ=n\tan \theta-\sin \theta=n, then
View answer and explanation
Correct option: D
m2−n2=4mnm^2-n^2=4 \sqrt{m n}
Option D (m2−n2=4mnm^2-n^2=4 \sqrt{m n}) is correct because by using the given equations, we can derive the relationship between m, n, tan θ, and sin θ.
313 Easy
313 of 394
Application +4 | -1
The value of cot⁡  54∘tan⁡  36∘  +  tan⁡  20∘cot⁡  70∘\frac{{\cot \,\,54^\circ }}{{\tan \,\,36^\circ }}\,\, + \,\,\frac{{\tan \,\,20^\circ }}{{\cot \,\,70^\circ }} is
View answer and explanation
Correct option: B
2
Option B (2) is correct because cot⁡ 54∘\cot \,54^\circ is equal to tan⁡ 36∘\tan \,36^\circ and tan⁡ 20∘\tan \,20^\circ is equal to cot⁡ 70∘\cot \,70^\circ, simplifying the expression to 1 + 1 = 2.
314 Medium
314 of 394
Application +4 | -1
If 2cos⁡A=x+1/x2cos⁡B=y+1/y2 \cos A=x+1 / x 2 \cos B=y+1 / y then find the value of &#39; kk &#39; such that kcos⁡(A−B)=xy+yxk \cos (A-B)=\frac{x}{y}+\frac{y}{x}.
View answer and explanation
Correct option: A
2
Option A (2) is correct because by expressing x and y in terms of cosine and sine functions and substituting into the given expression, we can simplify it to 2cos⁡(A−B)2\cos(A-B), thus k=2.
315 Easy
315 of 394
Application +4 | -1
If fk(x)=1/k(sin⁡kx+cos⁡kx), {f_k}(x) = 1/k({\sin ^k}x + {\cos ^k}x),\, where x∈Rx \in R and k≥1,k \ge 1, then f4(x)−f6(x) {f_4}(x) - {f_6}(x)\, is equal to
View answer and explanation
Correct option: D
112\frac{1}{12}
Option D (1/12) is correct because, after substituting the given function and simplifying using trigonometric identities, the expression reduces to 1/12.

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316 Easy
316 of 394
Application +4 | -1
If sec⁡  θ+tan⁡  θ=k,  cos⁡  θ\sec \,\,\theta + \tan \,\,\theta = k,\,\,\cos \,\,\theta equals to
View answer and explanation
Correct option: B
2kk2+1\frac{{2k}}{{{k^2} + 1}}
Option C (815\frac{8}{15}) is correct because by using the area formula and Heron's formula, we can find the value of tan B.
317 Medium
317 of 394
Application +4 | -1
The sides of a △ABC\triangle A B C are in AP, if the angles AA and CC are the greatest and the smallest angles respectively, then 4(1−cos⁡A)(1−cos⁡C)4(1-\cos A)(1-\cos C) is equal to
View answer and explanation
Correct option: B
4cos⁡2A+C24 \cos ^2 \frac{A+C}{2}
Option B (4cos⁡2A+C24 \cos ^2 \frac{{A+C}}{{2}}) is correct because by using the properties of AP and trigonometric identities, the expression simplifies to this form.
318 Easy
318 of 394
Application +4 | -1
If 3sin⁡θ+4cos⁡θ=53 \sin \theta+4 \cos \theta=5, then the value of 4sin⁡θ−3cos⁡θ4 \sin \theta-3 \cos \theta is equal to
View answer and explanation
Correct option: D
0
Option D (0) is correct because, given 3sin(θ\theta)+4cos(θ\theta)=5, we can find the value of 4sin(θ\theta)-3cos(θ\theta) by using the identity (3sin(θ\theta)+4cos(θ\theta))2 + (4sin(θ\theta)-3cos(θ\theta))2 = 25, which results in 4sin(θ\theta)-3cos(θ\theta)=0.
319 Medium
319 of 394
Understanding +4 | -1
cos⁡4θ−sin⁡4θ\cos ^4 \theta-\sin ^4 \theta is equal to
View answer and explanation
Correct option: B
2cos⁡2θ−12 \cos ^2 \theta-1
Option B (2cos⁡2θ−12 \cos ^2 \theta-1) is correct because cos⁡4θ−sin⁡4θ\cos ^4 \theta - \sin ^4 \theta simplifies to cos⁡2θ\cos 2\theta, which is equal to 2cos⁡2θ−12 \cos ^2 \theta - 1.
320 Easy
320 of 394
Application +4 | -1
If acos⁡3α+3a  cos⁡α  sin⁡2=m  a{\cos ^3}\alpha + 3a\,\,\cos \alpha \,\,{\sin ^2} = m\; and a  sin⁡3α+3a  cos⁡2α  sin⁡α=n,   then  (m+n)2/3+  (m−n)2/3  a\,\,{\sin ^3}\alpha + 3a\,\,{\cos ^2}\alpha \,\,\sin \alpha = n,\;\,\mathrm{then}\;{\left( {m + n} \right)^{2/3}} + \;{\left( {m - n} \right)^{2/3}}\; is equal to
View answer and explanation
Correct option: C
2a2/32{a^{2/3}}
Option C (2a2/32{a^{2/3}}) is correct because simplifying the given expressions and using the binomial theorem leads to 2a2/32{a^{2/3}}.
321 Easy
321 of 394
Application +4 | -1
Minimum area of the triangle formed by any tangent to the ellipse x24+y22=1\frac{x^2}{4}+\frac{y^2}{2}=1 with coordinate axes is
View answer and explanation
Correct option: D
222 \sqrt{2}
Option D (222 \sqrt{2}) is correct because it represents the minimum area of the triangle formed by the tangent to the ellipse and the coordinate axes.
322 Easy
322 of 394
Application +4 | -1
tan⁡A1+sec⁡A+1+sec⁡Atan⁡A\frac{\tan A}{1+\sec A}+\frac{1+\sec A}{\tan A} is equal to
View answer and explanation
Correct option: C
2cosec⁡A2 \operatorname{cosec} A
Option C (2cosec⁡A2 \operatorname{cosec} A) is correct because simplifying the given expression using trigonometric identities leads to 2cosec⁡A2 \operatorname{cosec} A.
323 Medium
323 of 394
Analyzing +4 | -1
3  (sin⁡x−cos⁡x)4+6(sin⁡x+cos⁡x)2+4(sin⁡6x+cos⁡6x)3\;{\left( {\sin x - \cos x} \right)^4} + 6{\left( {\sin x + \cos x} \right)^2} + 4\left( {{{\sin }^6}x + {{\cos }^6}x} \right) is equal to
View answer and explanation
Correct option: B
13
Option B (13) is correct because by expanding and simplifying the given trigonometric expression, we find that it equals 13.
324 Medium
324 of 394
Application +4 | -1
cos⁡2A2+cos⁡2B2+cos⁡2C2=\cos ^2 \frac{A}{2}+\cos ^2 \frac{B}{2}+\cos ^2 \frac{C}{2}=
View answer and explanation
Correct option: B
2+2sin⁡A2sin⁡B2sin⁡C22+2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}
Option B (2+2sin⁡A2sin⁡B2sin⁡C22+2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}) is correct because by applying half-angle formulas and trigonometric identities, the expression simplifies to the given form.
325 Medium
325 of 394
Application +4 | -1
If α+β−γ=π\alpha+\beta-\gamma=\pi, then sin⁡2α+sin⁡2β−sin⁡2γ=\sin ^2 \alpha+\sin ^2 \beta-\sin ^2 \gamma=
View answer and explanation
Correct option: C
2sin⁡αsin⁡βcos⁡γ2 \sin \alpha \sin \beta \cos \gamma
Option A (2sin⁡αsin⁡βcos⁡γ2 \sin \alpha \sin \beta \cos \gamma) is correct because by using trigonometric identities and the given condition α+β−γ=π\alpha + \beta - \gamma = \pi, the expression simplifies to 2sin⁡αsin⁡βcos⁡γ2 \sin \alpha \sin \beta \cos \gamma.
326 Easy
326 of 394
Application +4 | -1
If sin⁡  x+sin⁡2x=1\sin \,\,x + {\sin ^2}x = 1, then value of cos⁡2 x+cos⁡4x{\cos ^2}\,x + {\cos ^4}x is
View answer and explanation
Correct option: A
1\mathrm{1}
Option A (1) is correct because given sin⁡ x+sin⁡2x=1\sin \,x + {\sin ^2}x = 1, we can deduce that sin⁡x=cos⁡2x\sin x = {\cos ^2}x, and substituting this into cos⁡2 x+cos⁡4x{\cos ^2}\,x + {\cos ^4}x gives us 1.
327 Medium
327 of 394
Application +4 | -1
If sin⁡(x+y)sin⁡(x−y)=a+ba−b\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}, then tan⁡xtan⁡y\frac{\tan x}{\tan y} is equal to
View answer and explanation
Correct option: B
ab\frac{a}{b}
Option B (ab\frac{{a}}{{b}}) is correct because by applying the componendo and dividendo rule and simplifying, the ratio tan⁡xtan⁡y\frac{{\tan x}}{{\tan y}} is found to be ab\frac{{a}}{{b}}.
328 Easy
328 of 394
Application +4 | -1
The value of tan⁡19π3\tan \frac{19 \pi}{3} is n\sqrt{n}. Value of &#39;nn&#39; is
View answer and explanation
Correct option: A
3
Option A (3) is correct because simplifying the given expression using trigonometric identities and the given value of \tan θ leads to 34\frac{{3}}{{4}}, which means m = 3.
329 Hard
329 of 394
Application +4 | -1
If cos⁡α=3/5,cos⁡β=4/5,α,β>0\cos \alpha=3 / 5, \cos \beta=4 / 5, \alpha, \beta>0 and are acute angles, then cos⁡α−β2\cos \frac{\alpha-\beta}{2} is
View answer and explanation
Correct option: D
None of these
The correct answer is None of these, as the calculated value of cos⁡α−β2\cos \frac{\alpha-\beta}{2} is 752\frac{7}{5 \sqrt{2}}, which is not among the given options.
330 Medium
330 of 394
Application +4 | -1
The value of ∑k=13  cos⁡2(2k−1)π12\sum \limits_{k = 1}^3 \,\,{\cos ^2}\left( {2k - 1} \right)\frac{\pi }{{12}}, is
View answer and explanation
Correct option: D
3/23/2
Option D (3/2) is correct because by substituting the values of k and using trigonometric identities, the summation simplifies to 32\frac{3}{2}.
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