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UPSC NDA I and II 2020 Math Solved Paper

Question Numbers: 52-54

Direction: Read the following information and answer the three items that follow:

Let α = β = 15°.

Question : 52 of 120

Marks: +1, -0

What is the value of sin α + cos β ?

Solution:

Concept:

Use the identities

\cos \theta = \sin(90^\circ - \theta)

and

\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}

Explanation:

We are given

\alpha = 15^\circ

and

\beta = 15^\circ

Compute

\sin 15^\circ + \cos 15^\circ

Rewrite

\cos 15^\circ = \sin(90^\circ - 15^\circ) = \sin 75^\circ

Now

\sin 15^\circ + \sin 75^\circ = 2\sin\frac{15+75}{2}\cos\frac{75-15}{2}

2\sin 45^\circ \cos 30^\circ

Substitute

\sin 45^\circ = \frac{1}{\sqrt{2}}

and

\cos 30^\circ = \frac{\sqrt{3}}{2}

Expression =

2 \times \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{\sqrt{2}}

Answer:

\frac{\sqrt{3}}{\sqrt{2}}

, which corresponds to option D.

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