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 · Easy

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Showing 151-180 of 218 questions

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151 Easy

151 of 218

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

152 Easy

152 of 218

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.

153 Easy

153 of 218

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.

154 Easy

154 of 218

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.

155 Easy

155 of 218

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.

156 Easy

156 of 218

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.

157 Easy

157 of 218

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

158 Easy

158 of 218

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.

159 Easy

159 of 218

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.

160 Easy

160 of 218

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.

161 Easy

161 of 218

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

162 Easy

162 of 218

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}}

163 Easy

163 of 218

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.

164 Easy

164 of 218

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

165 Easy

165 of 218

Application +4 | -1

\tan \mathrm{A}=\frac{1-\cos \mathrm{B}}{\sin \mathrm{B}}

then

\tan 2 \mathrm{~A}=

View answer and explanation

Correct option: A

\tan B

Option A (

\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 \frac{B}{2}

, which leads to

\tan 2A = \tan B

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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

166 Easy

166 of 218

Application +4 | -1

\mathrm{If}\,\,\;\mathrm{\theta }

is an acute angle and

\tan \,\,\mathrm{\theta } = \frac{1}{{\sqrt 7 }}

, then the value of

\frac{{\mathrm{cose}{\mathrm{c}^2}\,\mathrm{\theta } - {{\sec }^2}\,\,\mathrm{\theta }}}{{\mathrm{cose}{\mathrm{c}^2}\,\mathrm{\theta } + {{\sec }^2}\,\,\mathrm{\theta }}}

View answer and explanation

Correct option: A

3\mathrm{ }/4

Option A (

3\mathrm{ }/4

) is correct because it is obtained by substituting the given value of

\tan \theta

and using trigonometric identities to simplify the expression.

167 Easy

167 of 218

Understanding +4 | -1

Assertion: The second hand rotates through an angle of

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.

168 Easy

168 of 218

Application +4 | -1

\sin \,\,\theta + \cos \,\,\theta = \sqrt 2 \cos \,\,\theta

then

\cos \,\,\theta - \sin \,\,\theta

is equal to

View answer and explanation

Correct option: B

\sqrt 2 \,\,\sin \theta

Option B (

\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 \theta - \sin \theta

169 Easy

169 of 218

Application +4 | -1

The value of

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

View answer and explanation

Correct option: B

\tan \theta / 2

Option B (

\tan \theta / 2

) is correct because simplifying the given expression using half-angle formulas results in

\tan \theta / 2

170 Easy

170 of 218

Application +4 | -1

The value of

\frac{{\cot \,\,54^\circ }}{{\tan \,\,36^\circ }}\,\, + \,\,\frac{{\tan \,\,20^\circ }}{{\cot \,\,70^\circ }}

View answer and explanation

Correct option: B

Option B (2) is correct because

\cot \,54^\circ

is equal to

\tan \,36^\circ

and

\tan \,20^\circ

is equal to

\cot \,70^\circ

, simplifying the expression to 1 + 1 = 2.

171 Easy

171 of 218

Application +4 | -1

{f_k}(x) = 1/k({\sin ^k}x + {\cos ^k}x),\,

where

x \in R

and

k \ge 1,

then

{f_4}(x) - {f_6}(x)\,

is equal to

View answer and explanation

Correct option: D

\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.

172 Easy

172 of 218

Application +4 | -1

\sec \,\,\theta + \tan \,\,\theta = k,\,\,\cos \,\,\theta

equals to

View answer and explanation

Correct option: B

\frac{{2k}}{{{k^2} + 1}}

Option C (

\frac{8}{15}

) is correct because by using the area formula and Heron's formula, we can find the value of tan B.

173 Easy

173 of 218

Application +4 | -1

3 \sin \theta+4 \cos \theta=5

, then the value of

4 \sin \theta-3 \cos \theta

is equal to

View answer and explanation

Correct option: D

Option D (0) is correct because, given 3sin(

\theta

)+4cos(

\theta

)=5, we can find the value of 4sin(

\theta

)-3cos(

\theta

) by using the identity (3sin(

\theta

)+4cos(

\theta

))² + (4sin(

\theta

)-3cos(

\theta

))² = 25, which results in 4sin(

\theta

)-3cos(

\theta

)=0.

174 Easy

174 of 218

Application +4 | -1

a{\cos ^3}\alpha + 3a\,\,\cos \alpha \,\,{\sin ^2} = m\;

and

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

2{a^{2/3}}

Option C (

2{a^{2/3}}

) is correct because simplifying the given expressions and using the binomial theorem leads to

2{a^{2/3}}

175 Easy

175 of 218

Application +4 | -1

Minimum area of the triangle formed by any tangent to the ellipse

\frac{x^2}{4}+\frac{y^2}{2}=1

with coordinate axes is

View answer and explanation

Correct option: D

2 \sqrt{2}

Option D (

2 \sqrt{2}

) is correct because it represents the minimum area of the triangle formed by the tangent to the ellipse and the coordinate axes.

176 Easy

176 of 218

Application +4 | -1

\frac{\tan A}{1+\sec A}+\frac{1+\sec A}{\tan A}

is equal to

View answer and explanation

Correct option: C

2 \operatorname{cosec} A

Option C (

2 \operatorname{cosec} A

) is correct because simplifying the given expression using trigonometric identities leads to

2 \operatorname{cosec} A

177 Easy

177 of 218

Application +4 | -1

\sin \,\,x + {\sin ^2}x = 1

, then value of

{\cos ^2}\,x + {\cos ^4}x

View answer and explanation

Correct option: A

\mathrm{1}

Option A (1) is correct because given

\sin \,x + {\sin ^2}x = 1

, we can deduce that

\sin x = {\cos ^2}x

, and substituting this into

{\cos ^2}\,x + {\cos ^4}x

gives us 1.

178 Easy

178 of 218

Application +4 | -1

The value of

\tan \frac{19 \pi}{3}

\sqrt{n}

. Value of '

n

' is

View answer and explanation

Correct option: A

Option A (3) is correct because simplifying the given expression using trigonometric identities and the given value of \tan θ leads to

\frac{{3}}{{4}}

, which means m = 3.

179 Easy

179 of 218

Application +4 | -1

x^2+y^2=1 ; x

y \in R

, then

\left(3 x-4 x^2\right)^2

+\left(3 y-4 y^3\right)

is equal to

View answer and explanation

Correct option: A

1

Option A (1) is correct because by substituting

x = \cos t

and

y = \sin t

, the expression simplifies to

(\cos 3t)^2 + (\sin 3t)^2

, which equals 1.

180 Easy

180 of 218

Application +4 | -1

In a

\triangle A B C

, the sides

a, b

and

c

are such that they are the roots of

x^3-11 x^2+38 x-40=0

. Then,

\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}

is equal to

View answer and explanation

Correct option: C

\frac{9}{16}

Option C (

\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

\frac{9}{16}

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