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UPSC NDA I and II 2020 Math Solved Paper

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Question : 107 of 120

Marks: +1, -0

What is the maximum value of sin x ⋅ cos x ?

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

Solution:

Concept:

The product

\sin x \cdot \cos x

can be expressed using the double-angle identity

\sin 2x = 2 \sin x \cos x

.

The maximum value of

\sin \theta

(for any real

\theta

) is

1

.

Explanation:

Let

f(x) = \sin x \cdot \cos x

.

Multiply and divide by

2

:

f(x) = \frac{1}{2} \times (2 \sin x \cos x) = \frac{1}{2} \sin 2x

.

The function

\sin 2x

ranges from

-1

to

1

.

Hence its maximum value is

1

.

Therefore, the maximum value of

f(x)

is

\frac{1}{2} \times 1 = \frac{1}{2}

.

Answer:

\frac{1}{2}

(option C)

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