The value of $1 + \cos \,\,56^\circ + \cos \,\,58^\circ - \cos \,\,66^\circ $ is

$4\,\,\cos \,\,28^\circ \,\,\sin \,\,29^\circ \,\,\cos \,\,33^\circ $

$4\,\,\cos \,\,28^\circ \,\,\sin \,\,29^\circ \,\,\sin \,\,33^\circ $

Which of the following functions have period $2{\rm{\pi }}$?

$y = \sin \left( {2\pi t + \frac{\pi }{3}} \right) + 2\sin \left( {3\pi t + \frac{\pi }{4}} \right) + 3\sin 5\pi t$

$y = \sin \frac{\pi }{3}t + \sin \frac{\pi }{4}t$

The value of $\frac{{\sin \,\,\left( {B + A} \right) + \;{\rm{cos}}\,\,\left( {B - A} \right)}}{{\sin \,\,\left( {B - A} \right) + \cos \,\,\left( {B + A} \right)}}$ is equal to

$\frac{{\cos \,\,B + \sin B}}{{\cos \,\,B - \sin \,\,B}}$

If $A + B + C\; = \frac{{3\,\pi }}{2}$, then $\cos \,\,2A + \cos \,\,2B + \cos \,\,2C$ is equal to

$1 + 2\cos A\,\,\cos B\,\,\cos C$

If $A + B + C = \pi $, then $\sin \,\,2A + \sin \,\,2B + \sin \,\,2C$ is equal to

$2\,\,\cos A\,\,\cos B\,\,\cos C$

$2\,\,\sin A\,\,\;{\rm{sin}}\;B\,\,\sin C$

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JEE • MATHEMATICS • TRANSFORMATIONS

Transformations practice · Medium

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Practice format and chapter quiz

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 31-45 of 45 questions

31 Medium

31 of 45

Application +4 | -1

Given

\frac{\pi}{2} < \alpha < \pi

, then the expression

\sqrt{\frac{1-\sin \alpha}{1+\sin \alpha}}+\sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}

is equal to

View answer and explanation

Correct option: B

-\frac{2}{\cos \alpha}

Option B (

-\frac{2}{\cos \alpha}

) is correct because, given the range of

\alpha

, the expression simplifies to this value using trigonometric identities.

32 Medium

32 of 45

Application +4 | -1

A + B + C = 270^\circ

, then

\cos \,\,2A + \cos \,\,2B + \cos \,\,2C

is equal to

View answer and explanation

Correct option: C

1 - 4\,\,\sin A\,\,\sin B\,\,\sin C\;

Option D (

1 - 4\,\cos A\,\cos B\,\cos C

) is correct because it is the result of applying trigonometric identities to simplify the given expression with the condition

A + B + C = 270^\circ

33 Medium

33 of 45

Analyzing +4 | -1

\cos \,\,\left( {\alpha + \beta } \right)\sin \,\,\left( {\gamma + \delta } \right) = \cos \,\,\left( {\alpha - \beta } \right)\sin \,\,\left( {\gamma - \delta } \right)

, then

\cot \alpha \,\,\cot \beta \,\,\cot \gamma

is equal to

View answer and explanation

Correct option: A

\cot \,\,\delta

Option A (

\cot \delta

) is correct because after applying trigonometric identities and simplifying the given equation, we arrive at the relation

\cot \alpha \cot \beta \cot \gamma = \cot \delta

34 Medium

34 of 45

Application +4 | -1

\cos \,\,\left( {\theta + \phi } \right) = m\,\,\cos \,\,\left( {\theta - \phi } \right)

is equal to

View answer and explanation

Correct option: C

\frac{{1 - m}}{{1 + m}}\,\,\cot \,\,\phi

Option C (

\frac{{1 - m}}{{1 + m}}\cot \phi

) is correct because it is the result of applying the componendo and dividendo rule and trigonometric identities to the given equation.

35 Medium

35 of 45

Application +4 | -1

- \,\pi \le x \le \pi ,\;\, - \,\pi \le y \le \pi \;\,\mathrm{and}\,\,\cos \,\,x + \cos \,\,y = 2

, then the value of

\mathrm{cos}\left( {x - y} \right)

View answer and explanation

Correct option: C

Option C (1) is correct because the maximum value of cosine is 1, so both cos(x) and cos(y) must be 1. This implies x = 0 and y = 0, therefore cos(x - y) = cos(0) = 1.

36 Medium

36 of 45

Application +4 | -1

The value of

1 + \cos \,\,56^\circ + \cos \,\,58^\circ - \cos \,\,66^\circ

View answer and explanation

Correct option: A

4\,\,\cos \,\,28^\circ \,\,\cos \,\,29^\circ \,\,\sin 33^\circ

Option A (

4\cos 28^\circ \cos 29^\circ \sin 33^\circ

) is correct because it represents the simplified form of the trigonometric expression after applying the sum-to-product and other trigonometric identities.

37 Medium

37 of 45

Analyzing +4 | -1

Which of the following functions have period

2\mathrm{\pi }

View answer and explanation

Correct option: C

y = \sin t + \cos 2t

Option C is correct. The period of

\sin t

2\pi

and the period of

\cos 2t

\pi

. The least common multiple of

2\pi

and

\pi

2\pi

38 Medium

38 of 45

Application +4 | -1

The value of

\frac{{\sin \,\,\left( {B + A} \right) + \;\mathrm{cos}\,\,\left( {B - A} \right)}}{{\sin \,\,\left( {B - A} \right) + \cos \,\,\left( {B + A} \right)}}

is equal to

View answer and explanation

Correct option: B

\frac{{\cos \,\,A + \sin \,\,A}}{{\cos \,\,A - \sin \,\,A}}

Option B (

\frac{{\cos \\,A + \sin \\,A}}{{\cos \\,A - \sin \\,A}}

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

\frac{{\cos \\,A + \sin \\,A}}{{\cos \\,A - \sin \\,A}}

39 Medium

39 of 45

Analyzing +4 | -1

In a

\triangle A B C, a, b

and

c

are the sides of the triangle opposite to the angles

2222222222 \mathrm{~d} A, B, C

, respectively. Then, the value of

a^3 \sin (B-C)+b^3 \sin (C-A)+c^3 \sin (A-B)

View answer and explanation

Correct option: A

Option A (0) is correct because the expression simplifies to zero using trigonometric identities and the sine rule.

40 Medium

40 of 45

Application +4 | -1

f:R \to R

and is defined by

f\left( x \right) = \frac{1}{{2 - \cos 3x}}

for each

x \in R

, then the range

f

View answer and explanation

Correct option: B

\left[ {1/3,\mathrm{ }1} \right]

Option B

\left[ {1/3,\mathrm{ }1} \right]

is correct because the range is found by considering the bounds of the cosine function and its effect on the given function.

41 Medium

41 of 45

Application +4 | -1

A + B + C\; = \frac{{3\,\pi }}{2}

, then

\cos \,\,2A + \cos \,\,2B + \cos \,\,2C

is equal to

View answer and explanation

Correct option: D

1 - 4\sin A\,\,\sin B\,\,\sin C

Option D (

1 - 4\sin A\sin B\sin C

) is correct because, with the given condition, the expression simplifies to this form using trigonometric identities.

42 Medium

42 of 45

Analyzing +4 | -1

y = {\cos ^2}x + {\sec ^2}x

, then

View answer and explanation

Correct option: C

y \ge 2

Option C (

y \ge 2

) is correct because

{\cos ^2}x + {\sec ^2}x

can be rewritten as

(\cos x - \sec x)^2 + 2

, which is always greater than or equal to 2.

43 Medium

43 of 45

Analyzing +4 | -1

\;A + B + C = \pi \,\,\;\mathrm{and}\,\,\cos \,\,A = \cos \,\,B\,\,\cos \,\,C

then

\tan \,\,B\,\,\tan \,\,C

is equal to

View answer and explanation

Correct option: B

\mathrm{1}

Option B (1) is correct because by using the given condition

A + B + C = \pi

and the trigonometric identity, we can derive that

\tan B \tan C = 1

44 Medium

44 of 45

Analyzing +4 | -1

If the equation

\sin \,\,\theta \left( {\sin \,\,\theta + 2\,\,\cos \,\,\theta } \right) = a

has a real solution, then the shortest interval containing

'a'

View answer and explanation

Correct option: A

\left[ {\frac{{1 - \sqrt 5 }}{2},\,\,\frac{{1 + \sqrt 5 }}{2}} \right]

Option A is correct because the range of 'a' is determined by finding the maximum and minimum values of the trigonometric expression.

45 Medium

45 of 45

Application +4 | -1

A + B + C = \pi

, then

\sin \,\,2A + \sin \,\,2B + \sin \,\,2C

is equal to

View answer and explanation

Correct option: A

4\,\,\sin A\,\,\sin B\,\,\sin C

Option A (

4\,\,\sin A\,\,\sin B\,\,\sin C

) is correct because it is the result of applying trigonometric identities to simplify the expression

\sin 2A + \sin 2B + \sin 2C

when

A + B + C = \pi

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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

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