If $x = h + a\,\,\sec \,\,\theta $ and $y = k + b\,{\rm{cosec}}\;\,\theta $. Then,

$\frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}}\,\, + \,\,\frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1$

$\frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}\;}}\,\, - \,\,\frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1$

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JEE • MATHEMATICS • TRIGONOMETRY RATIOS

Trigonometry ratios practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 271-300 of 394 questions

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271 Easy

271 of 394

Application +4 | -1

The maximum and minimum value of

f(x)

=a b \sin x+b \sqrt{1-a^2} \cos x+c

lie in the interval ( assuming

|a|<1, b>0)

View answer and explanation

Correct option: C

[c-b, b+c]

Option C ([c-b, b+c]) is correct because the maximum and minimum values of the function lie in the interval [c-b, b+c], which is found by rewriting the function as a sine function and using its amplitude.

272 Easy

272 of 394

Application +4 | -1

0 < x < \pi

, and

\sin x+\cos x=1 / 3

, then a possible value of

\tan x

View answer and explanation

Correct option: A

\frac{1}{8}(-9+\sqrt{17})

The correct answer is A (

\frac{1}{8}(-9+\sqrt{17})

) because by squaring the given equation and using trigonometric identities, we can solve for tan x.

273 Easy

273 of 394

Application +4 | -1

\cos ^2 48^{\circ}-\sin ^2 12^{\circ}=

View answer and explanation

Correct option: B

\frac{\sqrt{5}+1}{8}

Option B (

\frac{{\sqrt{5}+1}}{{8}}

) is correct because using the identity

\cos^2 A - \sin^2 B = \cos(A+B)\cos(A-B)

, the expression simplifies to

\frac{{\sqrt{5}+1}}{{8}}

274 Easy

274 of 394

Application +4 | -1

15 \sin ^4 x+10 \cos ^4 x=6

, Then

\tan ^2 x=

View answer and explanation

Correct option: C

2 / 3

Option C (

2 / 3

) is correct because by using the identity

\sin^2 x + \cos^2 x = 1

and simplifying the given equation, we can find that

\tan^2 x = 2/3

275 Easy

275 of 394

Analyzing +4 | -1

y = \tan x\cot 3x,x \in R,

then

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the range of y is

y \le \frac{1}{3}

y > 3

, which does not match any of the given options.

276 Medium

276 of 394

Application +4 | -1

\frac{x}{\cos \theta}=\frac{y}{\cos \left(\theta+\frac{2 \pi}{3}\right)}=\frac{z}{\cos \left(\theta-\frac{2 \pi}{3}\right)}

, then

x+y+z

is equal to

View answer and explanation

Correct option: C

Option C (0) is correct because using the given condition and the cosine addition formula, the expression simplifies to zero.

277 Medium

277 of 394

Analyzing +4 | -1

\sin A+\sin B+\sin C=3

, then

\cos A+\cos B+\cos C

is equal to

View answer and explanation

Correct option: D

Option D is correct because if the sum of sines is 3, each sine must be 1, making each angle

\frac{\pi}{2}

, thus each cosine is 0.

278 Medium

278 of 394

Application +4 | -1

In a

\triangle A B C

, if

a, b

and

c

are in AP, then the value of

\frac{\sin \frac{A}{2} \sin \frac{C}{2}}{\sin \frac{B}{2}}

View answer and explanation

Correct option: B

\frac{1}{2}

Option B (

\frac{1}{2}

) is correct because when

a, b, c

are in A.P., we can express

s

in terms of

b

, and then simplify the given expression using the properties of sine and the semi-perimeter.

279 Easy

279 of 394

Application +4 | -1

\sin x=\frac{-2 \sqrt{6}}{5}

and

x

lies in III quadrant, then the value of

\cot x

\frac{1}{m \sqrt{6}}

. Value of

m

View answer and explanation

Correct option: A

Option A (6) is correct because evaluating the expression using the values of trigonometric functions at the specified angles results in 6.

280 Easy

280 of 394

Application +4 | -1

x = h + a\,\,\sec \,\,\theta

and

y = k + b\,\mathrm{cosec}\;\,\theta

. Then,

View answer and explanation

Correct option: B

\frac{{{a^2}}}{{{{\left( {x - h} \right)}^2}}}\,\, + \,\,\frac{{{b^2}}}{{{{\left( {y - k} \right)}^2}}} = 1

Option B (

\frac{{{a^2}}}{{{{\left( {x - h} \right)}^2}}}\, + \,\frac{{{b^2}}}{{{{\left( {y - k} \right)}^2}}} = 1

) is correct because it is derived from the trigonometric identity

{{\cos }^2}\,\theta + {\sin ^2}\,\theta = 1

using the given parametric equations.

281 Easy

281 of 394

Application +4 | -1

The value of

\sin 765^{\circ}

\frac{1}{\sqrt{n}}

. Value of

n

View answer and explanation

Correct option: A

Option A (2) is correct because \sin 765^° simplifies to \sin 45^° which is \frac{{1}}{{

\sqrt{{2}}

}}, therefore n = 2.

282 Easy

282 of 394

Application +4 | -1

\tan 203^{\circ}+\tan 22^{\circ}+\tan 203^{\circ} \tan 22^{\circ}=

View answer and explanation

Correct option: C

Option C (1) is correct because by using the tangent addition formula and simplifying, the expression equals 1.

283 Medium

283 of 394

Application +4 | -1

\sin x+\sin ^2 x+\sin ^3 x=1

then evaluate the value of

\cos ^6 x-4 \cos ^4 x+8 \cos ^2 x

View answer and explanation

Correct option: A

Option A (4) is correct because by substituting

\sin x + \sin^2 x + \sin^3 x = 1

and using trigonometric identities, the expression

\cos^6 x - 4 \cos^4 x + 8 \cos^2 x

simplifies to 4.

284 Easy

284 of 394

Application +4 | -1

12 \cot ^2 \theta-31 \operatorname{cosec} \theta+32=0

, then the value of

\sin \theta

View answer and explanation

Correct option: C

\frac{4}{5}

\frac{3}{4}

Option C (

\frac{4}{5}

\frac{3}{4}

) is correct because it is obtained by solving the trigonometric equation for

\operatorname{cosec} \theta

and then finding the corresponding values of

\sin \theta

285 Medium

285 of 394

Application +4 | -1

x, y, z

are in A.P. then

\frac{\sin x-\sin z}{\cos z-\cos x}

is equal to -

View answer and explanation

Correct option: B

\cot y

Option B (

\cot y

) is correct because using the A.P. property and trigonometric identities simplifies the expression to

\cot y

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

286 Easy

286 of 394

Analyzing +4 | -1

For any

\theta \in \left( {\frac{\pi }{4},\frac{\pi }{2}} \right),

the expression

3{(\sin \theta - \cos \theta )^4} + 6{(\sin \theta + \cos \theta )^2} + 4{\sin ^6}\theta

equals

View answer and explanation

Correct option: D

13 - 4\,{\cos ^6}\theta

Option D (13 - 4cos⁶(θ)) is correct because after simplification using trigonometric identities, the expression reduces to this form.

287 Medium

287 of 394

Application +4 | -1

\frac{\cos 12^{\circ}-\sin 12^{\circ}}{\cos 12^{\circ}+\sin 12^{\circ}}+\frac{\sin 147^{\circ}}{\cos 147^{\circ}}=

View answer and explanation

Correct option: C

Option C (0) is correct because after simplifying using trigonometric identities, the expression equals 0.

288 Easy

288 of 394

Application +4 | -1

The value of

3 \sin \frac{\pi}{6} \sec \frac{\pi}{3}-4 \sin \frac{5 \pi}{6} \cot \frac{\pi}{4}

is equal to

View answer and explanation

Correct option: A

Option A (1) is correct because the expression simplifies to 1 after substituting trigonometric values and simplifying.

289 Easy

289 of 394

Application +4 | -1

\theta

lies in the first quadrant and

5\,\,\tan \,\,\theta = 4

, then

\frac{{5\,\,\sin \,\,\theta - 3\,\,\cos \,\,\theta }}{{\sin \,\,\theta + 2\,\,\cos \,\,\theta }}

is equal to

View answer and explanation

Correct option: A

\frac{5}{{14}}

Option A (5/14) is correct because after finding sin θ and cos θ from the given tan θ, substituting them into the expression yields 5/14.

290 Medium

290 of 394

Analyzing +4 | -1

In a

\triangle A B C, \tan A \cdot \tan B \cdot \tan C=9

. For such triangles, if

\tan ^2 A+\tan ^2 B+\tan ^2 C=k

, then

View answer and explanation

Correct option: B

k>27

Option B (

k>27

) is correct because using the AM-GM inequality and the fact that

\tan A + \tan B + \tan C = \tan A \tan B \tan C

in a triangle, we can show that

\tan^2 A + \tan^2 B + \tan^2 C > 27

291 Easy

291 of 394

Application +4 | -1

In a right angled triangle, the hypotenuse is four times as long as the perpendicular drawn to it from the opposite vertex. One of the acute angle is

View answer and explanation

Correct option: A

15^\circ

Option A (

15^\circ

) is correct because by setting up the geometric relationships in the right-angled triangle and using trigonometric ratios, we can derive that one of the acute angles is

15^\circ

292 Easy

292 of 394

Application +4 | -1

\sec \,\,\theta = x + \frac{1}{{4x}}

, then

\sec \,\,\theta + \tan \,\,\theta =

View answer and explanation

Correct option: B

2x,\,\,\frac{1}{{2x}}

Option B (

2x,\,\frac{1}{{2x}}

) is correct because by using the identity

\sec^2 \theta - \tan^2 \theta = 1

and solving the resulting equations, we find that

\sec \theta + \tan \theta

can be either

2x

\frac{1}{{2x}}

293 Medium

293 of 394

Application +4 | -1

x=h+a \sec \theta

and

y=k+b \operatorname{cosec} \theta

. Then,

View answer and explanation

Correct option: B

\frac{a^2}{(x-h)^2}+\frac{b^2}{(y-k)^2}=1

Option A (0) is correct because the given conditions imply that the determinant is a skew-symmetric matrix of order 3, which is always 0.

294 Medium

294 of 394

Application +4 | -1

If in a

\triangle A B C, \sin ^2 A+\sin ^2 B+\sin ^2 C=2

, then the triangle is always

View answer and explanation

Correct option: B

right angled

Option B (right angled) is correct because the given condition

\sin^2 A + \sin^2 B + \sin^2 C = 2

implies that one of the angles must be

90^\circ

, making the triangle right-angled.

295 Easy

295 of 394

Application +4 | -1

A

lies in the third quadrant and

3\tan A - 4 = 0

, then

5\sin \,\,2A + 3\sin A + 4\cos A =

View answer and explanation

Correct option: A

Option A (0) is correct because when substituting the given values into the expression, the terms cancel out, resulting in zero.

296 Medium

296 of 394

Application +4 | -1

If a

=\pi / 18

rad, then

\cos a+\cos 2 a+\ldots .+\cos 18 a

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because the sum telescopes to 0 due to the symmetry around

\frac{\pi}{2}

297 Medium

297 of 394

Application +4 | -1

The equation

a\,\,\cos \,\,\theta + b\,\,\sin \,\,\theta = c

has a solution, when

a,\;\,b

and

c

are real numbers such that

View answer and explanation

Correct option: C

{c^2} \le {a^2} + {b^2}

Option C (

{c^2} \le {a^2} + {b^2}

) is correct because the equation

a\cos \theta + b\sin \theta = c

has a solution only if

c

lies within the range

[-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}]

, which implies

{c^2} \le {a^2} + {b^2}

298 Easy

298 of 394

Application +4 | -1

a = \frac{\pi }{{18}}\,\,\mathrm{rad}

, then

\cos \,\,a + \cos \,\,2a + ... + \cos \,\,18\,a

is equal to

View answer and explanation

Correct option: B

- 1

Option B (

n\pi + {\left( { - 1} \right)^n}\frac{\pi }{4} - \frac{\pi }{4}

) is correct because it is the general solution to the trigonometric equation

\sin \theta + \cos \theta = 1

299 Medium

299 of 394

Application +4 | -1

Value of

2\left(\sin ^{6} \theta+\cos ^{6} \theta\right)

-3\left(\sin ^{4} \theta+\cos ^{4} \theta\right)+1

is :

View answer and explanation

Correct option: B

Option B (0) is correct because by using trigonometric identities and simplifying, the expression evaluates to 0.

300 Medium

300 of 394

Analyzing +4 | -1

\theta = \frac{{2\sin x}}{{1 + \sin x + \cos x}}

, then

\frac{{1 + \sin x - \cos x}}{{1 + \sin x}}

equals

View answer and explanation

Correct option: C

\theta

Option C (

\theta

) is correct because by using trigonometric identities, the expression can be simplified to be equal to the given value of

\theta

Revision notes and key takeaways

Review TRIGONOMETRY RATIOS in your textbook or class notes, then use this set to check your understanding.

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