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UPSC NDA I 2018 Mathematics Solved Paper

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Question : 58 of 120

Marks: +1, -0

What is sin 105° + cos 105° equal to?

sin 50°
cos 50°
$1/\sqrt{2}$
0

Solution:

Concept:

Use trigonometric identities to simplify the sum

sin 105° + cos 105°

into a standard value.

Explanation:

Step 1: Rewrite

cos 105°

using

cos (90° + 15°) = -sin 15°

.

So,

sin 105° + cos 105° = sin 105° + (-sin 15°) = sin 105° - sin 15°

.

Step 2: Apply the identity

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

.

Here,

A = 105°

,

B = 15°

.

Then,

sin 105° - sin 15° = 2 \, cos \frac{105°+15°}{2} \, sin \frac{105°-15°}{2}

.

Step 3: Simplify the angles:

\frac{105+15}{2} = \frac{120}{2} = 60°

and

\frac{105-15}{2} = \frac{90}{2} = 45°

.

So,

= 2 \, cos 60° \, sin 45°

.

Step 4: Evaluate the trigonometric values:

cos 60° = \frac{1}{2}

,

sin 45° = \frac{1}{\sqrt{2}}

.

Thus,

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

.

Answer:

sin 105° + cos 105° = \frac{1}{\sqrt{2}}

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