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JEE • MATHEMATICS • TRIGONOMETRY RATIOS

Trigonometry ratios practice · Mixed difficulty

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Showing 331-360 of 394 questions

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331 Medium
331 of 394
Application +4 | -1
If x+y−z+xyz=0x+y-z+x y z=0, then 2x1−x2+2y1−y2−2z1−z2\frac{2 x}{1-x^2}+\frac{2 y}{1-y^2}-\frac{2 z}{1-z^2} is equal to
View answer and explanation
Correct option: C
8xyz[(1−x2)(1−y2)(1−z2)]\frac{8 x y z}{\left[\left(1-x^2\right)\left(1-y^2\right)\left(1-z^2\right)\right]}
Option C (8xyz[(1−x2)(1−y2)(1−z2)]\frac{{8 x y z}}{{\left[(1-x^2)(1-y^2)(1-z^2)\right]}}) is correct because by substituting x=tan⁡Ax = \tan A, y=tan⁡By = \tan B, and −z=tan⁡C-z = \tan C and using trigonometric identities, the expression simplifies to the given form.
332 Medium
332 of 394
Application +4 | -1
The numerical value of sin⁡π18⋅sin⁡5π18⋅sin⁡7π18\sin \frac{\pi}{18} \cdot \sin \frac{5 \pi}{18} \cdot \sin \frac{7 \pi}{18} is equal to
View answer and explanation
Correct option: B
1/81 / 8
Option B (1/81 / 8) is correct because the given expression can be simplified using the identity sin⁡x⋅sin⁡(60∘−x)⋅sin⁡(60∘+x)=14sin⁡3x\sin x \cdot \sin (60^\circ - x) \cdot \sin (60^\circ + x) = \frac{{1}}{{4}} \sin 3x, leading to the solution 1/81/8.
333 Medium
333 of 394
Application +4 | -1
If cos⁡α=2cos⁡β−12−cos⁡β(0<α<β<π)\cos \alpha=\frac{2 \cos \beta-1}{2-\cos \beta}(0 < \alpha < \beta < \pi), then tan⁡α/2tan⁡β/2\frac{\tan \alpha / 2}{\tan \beta / 2} is equal to
View answer and explanation
Correct option: C
3\sqrt{3}
Option C (3\sqrt{3}) is correct because after applying trigonometric identities and simplifying, the ratio tan⁡α2tan⁡β2\frac{{\tan \frac{{\alpha}}{{2}}}}{{\tan \frac{{\beta}}{{2}}}} equals 3\sqrt{3}.
334 Medium
334 of 394
Application +4 | -1
Find the value of cot⁡5∘cot⁡10∘…..cot⁡85∘\cot 5^{\circ} \cot 10^{\circ} \ldots . . \cot 85^{\circ}.
View answer and explanation
Correct option: A
1
Option A (1) is correct because pairing the cotangents with their complementary angles results in a product of 1.
335 Easy
335 of 394
Application +4 | -1
If x2+y2=1;xx^2+y^2=1 ; x, y∈Ry \in R, then (3x−4x2)2\left(3 x-4 x^2\right)^2 +(3y−4y3)+\left(3 y-4 y^3\right) is equal to
View answer and explanation
Correct option: A
11
Option A (1) is correct because by substituting x=cos⁡tx = \cos t and y=sin⁡ty = \sin t, the expression simplifies to (cos⁡3t)2+(sin⁡3t)2(\cos 3t)^2 + (\sin 3t)^2, which equals 1.
336 Easy
336 of 394
Application +4 | -1
In a △ABC\triangle A B C, the sides a,ba, b and cc are such that they are the roots of x3−11x2+38x−40=0x^3-11 x^2+38 x-40=0. Then, cos⁡Aa+cos⁡Bb+cos⁡Cc\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c} is equal to
View answer and explanation
Correct option: C
916\frac{9}{16}
Option C (916\frac{9}{16}) is correct because by using Vieta's formulas to find the sum and sum of products of the roots, and then applying the cosine rule, the expression simplifies to 916\frac{9}{16}.
337 Easy
337 of 394
Application +4 | -1
The value of 2cos⁡20∘cos⁡40∘cos⁡80∘2 \cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ} is equal to ___.
View answer and explanation
Correct option: A
0.250.25
Option A (0.25) is correct because the expression simplifies to 1/4 using the double angle formula for sine repeatedly.
338 Easy
338 of 394
Understanding +4 | -1
Assertion: The measure of rotation of a given ray about its initial point is called an angle.

Reason: The point of rotation is called a vertex.
View answer and explanation
Correct option: B
Assertion is correct, reason is correct; reason is not a correct explanation for assertion
Assertion is correct, reason is correct; reason is not a correct explanation for assertion because while both statements are true, the reason doesn't explain why the measure of rotation is called an angle.
339 Easy
339 of 394
Analyzing +4 | -1
The angle θ\theta whose cosine equals to its tangent is given by
View answer and explanation
Correct option: C
sin⁡  θ=2  sin⁡  18∘\sin \,\,\theta = 2\,\,\sin \,\,18^\circ
Option C (sin⁡ θ=2 sin⁡ 18∘\sin \,\theta = 2\,\sin \,18^\circ) is correct because by setting cos⁡θ=tan⁡θ\cos \theta = \tan \theta and solving for sin⁡θ\sin \theta, we find that sin⁡θ\sin \theta is equal to 2sin⁡18∘2\sin 18^\circ.
340 Easy
340 of 394
Application +4 | -1
In △ABC,cos⁡2A+cos⁡2B−cos⁡2C=1\triangle A B C, \cos ^2 A+\cos ^2 B-\cos ^2 C=1, then
View answer and explanation
Correct option: C
∠C=π2\angle C=\frac{\pi}{2}
Option C (∠C=π2\angle C=\frac{\pi}{2}) is correct because the given equation can be rearranged to show that a2+b2=c2a^2 + b^2 = c^2, which implies that the triangle is right-angled at C.
341 Easy
341 of 394
Application +4 | -1
If tan⁡A=(1−cos⁡B)sin⁡B\tan \mathrm{A}=\frac{(1-\cos \mathrm{B})}{\sin \mathrm{B}}, then tan⁡2 A=……\tan 2 \mathrm{~A}=\ldots \ldots
View answer and explanation
Correct option: A
tan⁡B\tan \mathrm{B}
Option A (tan⁡B\tan B) is correct because simplifying the given expression for tan⁡A\tan A and then finding tan⁡2A\tan 2A using the double angle formula results in tan⁡B\tan B.
342 Easy
342 of 394
Application +4 | -1
If cos⁡  x−24  sin⁡x=  λcos⁡  (x+α),   0<α<π2\cos \,\,x - 24\,\,\sin x = \;\lambda \cos \,\,\left( {x + \alpha } \right),\,\;0 < \alpha < \frac{\pi }{2}, be true for all x  ∈Rx\; \in R, then

View answer and explanation
Correct option: A
λ=25\lambda = 25
Option A (λ=25\lambda = 25) is correct because by rewriting the given expression and comparing coefficients, we can find that λ=25\lambda = 25.
343 Easy
343 of 394
Analyzing +4 | -1
For −π2<θ<π2,  sin⁡θ+sin⁡2θ1+cos⁡θ+cos⁡2θ- \frac{\pi }{2} < \theta < \frac{\pi }{2},\,\,\frac{{\sin \theta + \sin 2\theta }}{{1 + \cos \theta + \cos 2\theta }} lies in the interval
View answer and explanation
Correct option: A
(−∞,   ∞)\left( { - \infty ,\;\,\infty } \right)
Option A is correct because the given expression simplifies to tan θ, and for -π/2 < θ < π/2, tan θ can take any real value.
344 Medium
344 of 394
Application +4 | -1
If α+β+γ=π\alpha+\beta+\gamma=\pi then the minimum value of cos⁡A+cos⁡B+cos⁡C\cos A+\cos B+\cos C
View answer and explanation
Correct option: D
is -3
Option D (is -3) is correct because the minimum value of cosine is -1, and if A=B=C=π, then cosA+cosB+cosC = -3.
345 Easy
345 of 394
Application +4 | -1
Suppose α,β>0\alpha, \beta>0 and α+2β=π/2\alpha+2 \beta=\pi / 2, then tan⁡(α+β)−2tan⁡α−tan⁡β\tan (\alpha+\beta)-2 \tan \alpha-\tan \beta is equal to:
View answer and explanation
Correct option: A
0
Option A (0) is correct because using the given condition α+2β=π2\alpha + 2\beta = \frac{\pi}{2}, we can show that tan⁡(α+β)−2tan⁡α−tan⁡β=0\tan (\alpha+\beta)-2 \tan \alpha-\tan \beta = 0.

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346 Easy
346 of 394
Application +4 | -1
If cos⁡(α+β)=4/5\cos (\alpha+\beta)=4 / 5 and sin⁡(α−β)=5/13\sin (\alpha-\beta)=5 / 13 where 0≤α,β≤π/40 \leq \alpha, \beta \leq \pi / 4, then tan⁡2α=\tan 2 \alpha=
View answer and explanation
Correct option: D
56/3356 / 33
Option D (56/3356 / 33) is correct because it is the result of applying the tangent addition formula after finding tan⁡(α+β)\tan(\alpha+\beta) and tan⁡(α−β)\tan(\alpha-\beta).
347 Medium
347 of 394
Analyzing +4 | -1
The expression (1+tan⁡  x+tan⁡2x) (1−cot⁡x+cot⁡2x)(1 + \tan \,\,x + {\tan ^2}x)\,(1 - \cot x + {\cot ^2}x) has the positive value for xx given by
View answer and explanation
Correct option: C
For all x∈R
Option C (For all x∈R) is correct because the given expression simplifies to a form that is always positive for all real values of x.
348 Easy
348 of 394
Application +4 | -1
If cos⁡θ=−35\cos \theta=\frac{-3}{5} and π<θ<3π2\pi < \theta < \frac{3 \pi}{2}, then the value of (cosec⁡θ+cot⁡θsec⁡θ−tan⁡θ)\left(\frac{\operatorname{cosec} \theta+\cot \theta}{\sec \theta-\tan \theta}\right) is equal to 1m\frac{1}{m}. Value of mm is
View answer and explanation
Correct option: A
6
Option A (1) is correct because simplifying the given expression using trigonometric identities results in 1.
349 Easy
349 of 394
Application +4 | -1
If θ=π/8\theta=\pi / 8, and f(θ)=cos⁡6θ+cos⁡63θ+f(\theta)=\cos ^6 \theta+\cos ^6 3 \theta+ cos⁡65θ+cos⁡67θ\cos ^6 5 \theta+\cos ^6 7 \theta then f(θ)f(\theta) is equal to:
View answer and explanation
Correct option: C
5/45 / 4
The correct answer is C (5/4) because by using trigonometric identities and substituting the value of θ, we can simplify the expression to obtain the result.
350 Easy
350 of 394
Application +4 | -1
The sum of the interior angles of a regular polygon is twice the sum of its exterior angles, the polygon is a
View answer and explanation
Correct option: A
Hexagon
Option A (Hexagon) is correct because the sum of interior angles being twice the sum of exterior angles leads to the equation (n−2)×180∘=2×360∘(n-2) \times 180^{\circ} = 2 \times 360^{\circ}, which solves to n=6.
351 Medium
351 of 394
Application +4 | -1
If x=sec⁡θ−tan⁡θ,y=cosec⁡θ+cot⁡θx=\sec \theta-\tan \theta, y=\operatorname{cosec} \theta+\cot \theta, then xy+1=….x y+1=\ldots ..
View answer and explanation
Correct option: C
y−xy-x
Option C (y−xy-x) is correct because simplifying the expression xy+1xy+1 using trigonometric identities results in y−xy-x.
352 Easy
352 of 394
Application +4 | -1
If π<θ<2π\pi < \theta < 2\pi, then 1+cos⁡  θ1−cos⁡  θ\sqrt {\frac{{1 + \cos \,\,\theta }}{{1 - \cos \,\,\theta }}} is equal to
View answer and explanation
Correct option: D
−cosec   θ−cot⁡  θ- \mathrm{cosec}\,\;\theta - \cot \,\,\theta
Option D (−cosec ;θ−cot⁡θ- \mathrm{cosec}\,\\;\theta - \cot \theta) is correct because, given π<θ<2π\pi < \theta < 2\pi, sin⁡θ\sin \theta is negative, which affects the sign when simplifying the square root.
353 Medium
353 of 394
Analyzing +4 | -1
If 0<x<π2,0 < x < \frac{\pi }{2},, then
View answer and explanation
Correct option: A
cos⁡(sin⁡x)>cos⁡x\cos \left( {\sin x} \right) > \cos x
Option A (cos⁡(sin⁡x)>cos⁡x\cos(\sin x) > \cos x) is correct because cos⁡x\cos x is a decreasing function on (0,π2)(0, \frac{\pi}{2}) and sin⁡x<x\sin x < x for 0<x<π20 < x < \frac{\pi}{2}.
354 Hard
354 of 394
Application +4 | -1
If α\alpha and β\beta are acute such that tan⁡(α+β)\tan (\alpha+\beta) and tan⁡(α−β)\tan (\alpha-\beta) satisfy the equation x2−4x+1=0x^2-4 x+1=0, then (α,β)=(\alpha, \beta)=
View answer and explanation
Correct option: C
(45∘,30∘)\left(45^{\circ}, 30^{\circ}\right)
Option C ((45°, 30°)) is correct because solving the quadratic equation and using the tangent addition and subtraction formulas, we find that α = 45° and β = 30°.
355 Medium
355 of 394
Application +4 | -1
If sin⁡(x−y)=cos⁡(x+y)=1/2\sin (x-y)=\cos (x+y)=1 / 2 then xx and yy lying between 0∘0^{\circ} and 90∘90^{\circ} are given by -
View answer and explanation
Correct option: D
x=45∘,y=15∘x=45^{\circ}, y=15^{\circ}
Option D (x=45∘,y=15∘x=45^{\circ}, y=15^{\circ}) is correct because solving the system of equations sin⁡(x−y)=1/2\sin (x-y) = 1/2 and cos⁡(x+y)=1/2\cos (x+y) = 1/2 yields these values for x and y within the given range.
356 Easy
356 of 394
Application +4 | -1
sin⁡70∘+cos⁡40∘cos⁡70∘+sin⁡40∘=\frac{\sin 70^{\circ}+\cos 40^{\circ}}{\cos 70^{\circ}+\sin 40^{\circ}}=
View answer and explanation
Correct option: C
3\sqrt{3}
Option C (3\sqrt{3}) is correct because after converting cosine to sine and applying sum-to-product formulas, the expression simplifies to 3\sqrt{3}.
357 Easy
357 of 394
Application +4 | -1
If sin⁡θ+sin⁡2θ+sin⁡3θ=sin⁡α\sin \theta+\sin 2 \theta+\sin 3 \theta=\sin \alpha and cos⁡θ\cos \theta +cos⁡2θ+cos⁡3θ=cos⁡α+\cos 2 \theta+\cos 3 \theta=\cos \alpha, then θ\theta is equal to
View answer and explanation
Correct option: A
α/2\alpha / 2
Option A (α/2\alpha / 2) is correct because dividing the sine equation by the cosine equation results in tan⁡2θ=tan⁡α\tan 2\theta = \tan \alpha, which implies θ=α/2\theta = \alpha / 2.
358 Medium
358 of 394
Application +4 | -1
In any triangle ABC,  ∑  sin⁡2A+sin⁡A+1sin⁡AABC,\,\,\sum \,\,\frac{{{{\sin }^2}A + \sin A + 1}}{{\sin A}} is always greater than
View answer and explanation
Correct option: A
9
Option A (9) is correct because each term in the summation is greater than or equal to 3, and there are three terms in the summation, resulting in a value greater than or equal to 9.
359 Medium
359 of 394
Application +4 | -1
In a quadrilateral, the measure of the angles are in A.P. If the ratio of the measures of the greatest and the smallest angles is 2:12: 1, then the measure of the greatest angle in radians is ...... .
View answer and explanation
Correct option: B
2π3\frac{2 \pi}{3}
Option B ([−1,12]\left[-1, \frac{{1}}{{\sqrt{{2}}}}\right]) is correct because solving the inequality sin⁡−1x≤cos⁡−1x\sin^{-1}x \le \cos^{-1}x leads to this interval.
360 Easy
360 of 394
Application +4 | -1
The radius of the circle whose arc of length 15 cm15 \mathrm{~cm} makes an angle of 3/43 / 4 radian at the centre is
View answer and explanation
Correct option: B
20 cm20 \mathrm{~cm}
Option B (20 cm20 \mathrm{~cm}) is correct because it is derived from the formula relating arc length, radius, and angle in radians: ℓ=rθ\ell = r\theta.
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