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UPSC CDS II 2024 Math Paper

Question Numbers: 56-57

Consider the following for the items that follow :

ABC is a triangle right-angled at B. Given that AC - A

B = 2

cm and B

C = 16

Question : 56 of 100

Marks: +1, -0

If ∠BAC = θ then what is sinθ + cosθ equal to?

Solution:

Concept:

Use Pythagoras theorem to find missing sides, then compute

\sin\theta

and

\cos\theta

Explanation:

Given:

\triangle ABC

is right‑angled at

B

AC - AB = 2\ \text{cm}

BC = 16\ \text{cm}

Let

AB = x

, so

AC = x + 2

Apply Pythagoras:

AC^2 = AB^2 + BC^2

(x+2)^2 = x^2 + 16^2

x^2 + 4x + 4 = x^2 + 256

4x = 252 \implies x = 63

Thus

AB = 63\ \text{cm}

AC = 65\ \text{cm}

BC = 16\ \text{cm}

Angle

\theta = \angle BAC

, so opposite side is

BC

and adjacent side is

AB

\sin\theta = \frac{BC}{AC} = \frac{16}{65}

\cos\theta = \frac{AB}{AC} = \frac{63}{65}

\sin\theta + \cos\theta = \frac{16}{65} + \frac{63}{65} = \frac{79}{65}

Answer:

\frac{79}{65}

(Option D)

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