If $\;A = {\sin ^2}{\rm{\theta }} + {\cos ^4}{\rm{\theta }}$ then for all real values of ${\rm{\theta }}$

$\frac{{13}}{{16}} \le A \le 1$

$\frac{3}{4} \le A \le \frac{{13}}{{16}}$

If $x=a \cos ^3 \theta, y=b \sin ^3 \theta$, then

$\left(\frac{a}{x}\right)^{2 / 3}+\left(\frac{b}{y}\right)^{2 / 3}=1$

$\left(\frac{b}{x}\right)^{2 / 3}+\left(\frac{a}{y}\right)^{2 / 3}=1$

$\left(\frac{x}{b}\right)^{2 / 3}+\left(\frac{y}{a}\right)^{2 / 3}=1$

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

$\frac{\sqrt{3}+1}{2 \sqrt{2}}$

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JEE • MATHEMATICS • TRIGONOMETRY RATIOS

Trigonometry ratios practice · Mixed difficulty

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Showing 31-60 of 394 questions

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31 Easy

31 of 394

Application +4 | -1

\sin \theta+\cos \theta=1

, then

\sin \theta \cos \theta=

View answer and explanation

Correct option: A

Option A (0) is correct because squaring the given equation and using the identity \sin² θ + \cos² θ = 1 leads to \sin θ \cos θ = 0.

32 Medium

32 of 394

Application +4 | -1

\frac{{\cos A}}{3} = \frac{{\cos B}}{4} = \frac{1}{5},\;\, - \frac{\pi }{2} < A < 0,\,\; - \frac{\pi }{2} < B < 0

then the value of

2\sin A + 4\sin B

View answer and explanation

Correct option: C

- 4

Option C (-4) is correct because after finding the values of sin A and sin B in the fourth quadrant, substituting them into the expression gives -4.

33 Medium

33 of 394

Application +4 | -1

The value of

\sin \,\,10^\circ + \sin \,\,20^\circ + \sin \,\,30^\circ + ... + \sin \,\,360^\circ

is equal to

View answer and explanation

Correct option: C

\sqrt 3

Option A (

\mathrm{0}

) is correct because the sum of sines of angles in arithmetic progression over a full cycle is zero.

34 Medium

34 of 394

Application +4 | -1

The value of

\cos ^2\left(\frac{3 \pi}{5}\right)+\cos ^2\left(\frac{4 \pi}{5}\right)

is equal to

View answer and explanation

Correct option: A

3 / 4

Option A (3/4) is correct because by converting the angles to degrees and using trigonometric identities, the expression simplifies to 3/4.

35 Easy

35 of 394

Application +4 | -1

Let

f(\theta)=\tan \left(\frac{\pi}{3}+\theta\right) \tan \left(\frac{\pi}{3}-\theta\right)

-\frac{\pi}{3}<\theta<\frac{\pi}{3}

, then

f\left(\frac{\pi}{12}\right)

is equal to:

View answer and explanation

Correct option: A

2+\sqrt{3}

The correct answer is A (

2+\sqrt{3}

) because by using trigonometric identities and substituting the value of

\theta

, we can simplify the expression to obtain the result.

36 Easy

36 of 394

Application +4 | -1

The value of

\cos 15^{\circ}-\sin 15^{\circ}

is equal to

View answer and explanation

Correct option: A

\frac{1}{\sqrt{2}}

Option A (

\frac{{1}}{{\sqrt{{2}}}}

) is correct because

\cos 15^{\circ} - \sin 15^{\circ}

can be simplified to

\sin 75^{\circ} - \sin 15^{\circ}

, which then simplifies to

\frac{{1}}{{\sqrt{{2}}}}

37 Easy

37 of 394

Application +4 | -1

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

, then

{\cos ^8}\,\,x + 2{\cos ^6}\,\,x + {\cos ^4}\,\,x =

View answer and explanation

Correct option: D

\mathrm{1}

Option D (1) is correct because by substituting

\sin x = \cos^2 x

into the expression

{\cos ^8}\\ x + 2{\cos ^6}\\ x + {\cos ^4}\\ x

, we can simplify it to

(\sin x + \sin^2 x)^2

, which equals 1.

38 Medium

38 of 394

Analyzing +4 | -1

\;A = {\sin ^2}\mathrm{\theta } + {\cos ^4}\mathrm{\theta }

then for all real values of

\mathrm{\theta }

View answer and explanation

Correct option: B

\frac{3}{4} \le A \le 1

Option B (

\frac{3}{4} \le A \le 1

) is correct because the minimum value of

A

\frac{3}{4}

and the maximum value is 1, which can be found by completing the square and using trigonometric properties.

39 Medium

39 of 394

Application +4 | -1

\tan \alpha=1 / 7

and

\sin \beta=1 / \sqrt{10}

where

0 < \alpha, \beta < \pi / 2

, then

2 \beta

is equal to

View answer and explanation

Correct option: A

\pi / 4-\alpha

Option A (

\pi / 4 - \alpha

) is correct because it represents the value of

2\beta

given the conditions

\tan \alpha = 1/7

and

\sin \beta = 1/\sqrt{{10}}

, derived using trigonometric identities.

40 Medium

40 of 394

Application +4 | -1

Interior angles of the regular dodecagon is

View answer and explanation

Correct option: D

5 \pi / 6

Option D (

5 \pi / 6

) is correct because the interior angle of a regular dodecagon (12 sides) is calculated using the formula

(n-2) \times 180^{\circ} / n

41 Easy

41 of 394

Application +4 | -1

0 < \mathrm{\theta } < \mathrm{\pi },

then the minimum value of

{\sin ^5}\theta + \mathrm{cose}{\mathrm{c}^5}\mathrm{\theta }

View answer and explanation

Correct option: C

Option C (2) is correct because the minimum value is found using the fact that

x + \frac{1}{x} \ge 2

for all

x > 0

, applied to the given expression.

42 Easy

42 of 394

Application +4 | -1

The value of

\left(1+\cos \frac{\pi}{6}\right)\left(1+\cos \frac{\pi}{3}\right)\left(1+\cos \frac{2 \pi}{3}\right)\left(1+\cos \frac{7 \pi}{6}\right)

\frac{m}{16}

. Value of

m

View answer and explanation

Correct option: A

Option A (6) is correct because given \cos θ =

\frac{{-3}}{{5}}

and θ in the third quadrant, calculating the expression results in

\frac{{1}}{{6}}

43 Easy

43 of 394

Application +4 | -1

The value of

\sin \,\,\left( {\pi + \theta } \right)\,\,\sin \,\,\left( {\pi - \theta } \right)\;\mathrm{cose}{\mathrm{c}^2}\,\theta

is equal to

View answer and explanation

Correct option: A

 - 1

Option A (- 1) is correct because the expression simplifies to -1 using trigonometric identities.

44 Medium

44 of 394

Application +4 | -1

(\sin \theta+\operatorname{cosec} \theta)^2+(\cos \theta+\sec \theta)^2

View answer and explanation

Correct option: A

\geq 9

Option A (

\geq 9

) is correct because the given expression can be simplified and shown to be greater than or equal to 9 using AM-GM inequality.

45 Easy

45 of 394

Application +4 | -1

\mathrm{\;}\sin x + \mathrm{cosec\;}x = 2\mathrm{\;\;}

then

{\sin ^n}x + \mathrm{cose}{\mathrm{c}^n}x\;

is equal to

View answer and explanation

Correct option: A

Option A (2) is correct because if sin x + cosec x = 2, then sin x = 1, and therefore sin^n x + cosec^n x = 1 + 1 = 2.

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46 Medium

46 of 394

Application +4 | -1

The maximum value of

a \cos x+b \sin x

View answer and explanation

Correct option: D

\left(a^2+b^2\right)^{1 / 2}

Option D (

\left(a^2+b^2\right)^{1 / 2}

) is correct because the maximum value of

a \cos x + b \sin x

\sqrt{{a^2 + b^2}}

47 Easy

47 of 394

Application +4 | -1

x=a \cos ^3 \theta, y=b \sin ^3 \theta

, then

View answer and explanation

Correct option: C

\left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{b}\right)^{2 / 3}=1

Option C (

\left(\frac{{x}}{{a}}\right)^{2 / 3}+\left(\frac{{y}}{{b}}\right)^{2 / 3}=1

) is correct because it is derived by expressing

\cos \theta

and

\sin \theta

in terms of x, y, a, and b, and then using the identity

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

48 Medium

48 of 394

Application +4 | -1

The resultant of three forces represented in magnitude and direction by the sides of a triangle

{ABC}

taken in order with

{BC}=5 {~cm}, {CA}=5 {~cm}

and

{AB}=8 {~cm}

, is a couple of moment

View answer and explanation

Correct option: B

24 units

Option B (24 units) is correct because the resultant of the forces is a couple with a moment equal to twice the area of the triangle.

49 Easy

49 of 394

Application +4 | -1

If in a triangle

A B C, 2 \cot B \cot C=1

, then

\cos B \cos C

will be equal to

View answer and explanation

Correct option: C

\cos A

Option C (

\cos A

) is correct because the given condition

2 \cot B \cot C = 1

can be manipulated using trigonometric identities to show that

\cos B \cos C = \cos A

50 Easy

50 of 394

Application +4 | -1

\mathrm{A,}\,\mathrm{B,}\,\mathrm{C,}\,\mathrm{D}

are the angles of a cyclic quadrilateral, then

\cos \,\,A + \cos \,\,B + \cos \,\,D\;

is equal to

View answer and explanation

Correct option: D

\mathrm{0}

Option D (0) is correct because in a cyclic quadrilateral, opposite angles are supplementary, leading to

\cos A + \cos C = 0

and

\cos B + \cos D = 0

, thus the sum is 0.

51 Easy

51 of 394

Application +4 | -1

\frac{\cos A}{3}=\frac{\cos B}{4}=\frac{1}{5}, \frac{-\pi}{2} < {A} < 0 , \frac{-\pi}{2} < {B} < 0

, then value of

2 \sin A+4 \sin B

-a

. The value of '

a

' is

View answer and explanation

Correct option: A

Option A (4) is correct because calculating 2\sin A + 4\sin B with the given conditions results in -4, and since it's equal to -a, a = 4.

52 Easy

52 of 394

Application +4 | -1

The value of the expression

{\sin ^6}\,\,\theta + {\cos ^6}\,\,\theta + 3{\sin ^2}\,\,\theta \,.\,\,{\cos ^2}\,\,\theta

equals

View answer and explanation

Correct option: D

\mathrm{1}

Option D (1) is correct because by expanding and simplifying the expression

{\sin ^6}\,{\theta } + {\cos ^6}\,{\theta } + 3{\sin ^2}\,{\theta }.\,{\cos ^2}\,{\theta }

, we can use the identity

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

to arrive at 1.

53 Hard

53 of 394

Application +4 | -1

The extreme values of

4\cos \left( {{x^2}} \right)\cos \left( {\frac{\mathrm{\pi }}{3} + {x^2}} \right)\cos \left( {\frac{\mathrm{\pi }}{3} - {x^2}} \right)

over

R

, are

View answer and explanation

Correct option: A

- 1,1

Option A (-1, 1) is correct because the extreme values are found by simplifying the trigonometric expression and determining the range of the resulting cosine function.

54 Easy

54 of 394

Application +4 | -1

\alpha+\beta=\frac{\pi}{2}

and

\beta+\gamma=\alpha

, then

\tan \alpha=

View answer and explanation

Correct option: B

\tan \beta+2 \tan \gamma

Option B (

\tan \beta+2 \tan \gamma

) is correct because it follows from trigonometric identities and algebraic manipulation of the given equations.

55 Medium

55 of 394

Application +4 | -1

In a triangle

\tan A+\tan B+\tan C=6

and

\tan A

\tan B=2

, then the values of

\tan A, \tan B

and

\tan C

are

View answer and explanation

Correct option: B

2,1,3

Option B (2,1,3) is correct because it satisfies the given conditions:

\tan A + \tan B + \tan C = 6

and

\tan A \tan B = 2

56 Medium

56 of 394

Application +4 | -1

\sin 2 \theta+\sin 2 \phi=1 / 2

and

\cos 2 \theta+\cos 2 \phi=3 / 2

, then

\cos ^2(\theta-\phi)=

View answer and explanation

Correct option: B

5 / 8

Option B (5 / 8) is correct because squaring and adding the given equations, then using trigonometric identities, leads to

\cos^2(\theta - \phi) = 5/8

57 Easy

57 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 it is derived from using the identity

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

and the value of

\cos 36^{\circ}

58 Easy

58 of 394

Application +4 | -1

\mathrm{\;}x\mathrm{\;}

lies in IInd quadrant, then

\;\frac{{\sqrt {1 + \sin \,\,x} + \sqrt {1 - \sin \,\,x} }}{{\sqrt {1 + \sin \,\,x} - \sqrt {1 - \sin \,\,x} }}\;

is equal to

View answer and explanation

Correct option: B

\tan \,\,\frac{x}{2}

Option B is correct because by simplifying the given expression using trigonometric identities and considering the quadrant of x, the result is

\tan \,\frac{x}{2}

59 Hard

59 of 394

Analyzing +4 | -1

The number of all possible triplets

\left( {{a_1},\,\,{a_2},\,{a_3}} \right)

such that

{a_1} + {a_2}\,\,\cos \,\,2x + {a_3}{\sin ^2}x = 0

for all

x

View answer and explanation

Correct option: D

None of these

The correct answer is None of these because there are infinitely many triplets that satisfy the given condition.

60 Medium

60 of 394

Application +4 | -1

\sum\limits_{{i=1}}^n \sin \theta_i=n

, then the value of

\sum\limits_{{i=1}}^n \cos \theta_i

View answer and explanation

Correct option: B

Option B (0) is correct because if

\sum_{i=1}^n \sin \theta_i = n

, then each

\sin \theta_i

must be 1, which means each

\theta_i = \frac{\pi}{2}

. Therefore, each

\cos \theta_i = 0

, and

\sum_{i=1}^n \cos \theta_i = 0

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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