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UPSC NDA 2 2016 Mathematics Solved Paper

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Question : 43 of 120

Marks: +1, -0

If

\sin \theta_{1} + \sin \theta_{2} + \sin \theta_{3} + \sin \theta_{4} = 4

, then what is the value of

\cos \theta_{1} + \cos \theta_{2} + \cos \theta_{3} + \cos \theta_{4}

?

0
1
2
4

Solution:

Concept:

The maximum value of

\sin \theta

is

1

and the maximum value of

\cos \theta

is

1

.

Explanation:

The given sum is

\sin \theta_{1} + \sin \theta_{2} + \sin \theta_{3} + \sin \theta_{4} = 4

.

Since each

\sin \theta

can be at most

1

, the only way the sum equals

4

is when each term is exactly

1

.

This means

\sin \theta_{1} = \sin \theta_{2} = \sin \theta_{3} = \sin \theta_{4} = 1

.

Therefore,

\theta_{1} = \theta_{2} = \theta_{3} = \theta_{4} = \frac{\pi}{2}

(principal value).

Now compute

\cos \theta_{1} + \cos \theta_{2} + \cos \theta_{3} + \cos \theta_{4}

:

\cos \frac{\pi}{2} = 0

, so each

\cos \theta

is

0

.

Thus, the sum is

0 + 0 + 0 + 0 = 0

.

Answer:

0

(Option A)

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