If 2 cot θ = 3, then find the value of \(\frac{\sqrt{13} \sin \theta-3 \tan \theta}{3 \tan \theta+\sqrt{13} \cos \theta}\).

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Question

This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 24 Jul 2023 Shift 1)

Answer 1

Given:

2cotθ = 3

Formula used:

Sinθ = P/H

Cosθ = B/H

Tanθ = P/B

Cotθ = B/P

H2 = P2 + B2

where P = Perpendicular

B = Base

H = Height

Calculation:

2cotθ = 3

⇒ cotθ = 3/2

on comparing with (Cotθ = B/P), we get

⇒ B = 3

⇒ P = 2

According to Pythagorean theorem

⇒ H² = P² + B²

⇒ H2 = 22 + 32

⇒ H = √13

⇒ Sinθ = 2/√13

⇒ Cosθ = 3/√13

⇒ Tanθ = 2/3

Putting all the values in the given equation:

\(\frac{\sqrt{13} \sin \theta-3 \tan \theta}{3 \tan \theta+\sqrt{13} \cos \theta}\)= (√13 * 2/√13 - 3 * 2/3 )/ (3 * 2/3 + √13 * 3/√13)

⇒ (2 - 2)/(2 + 3)

⇒ 0

The value of the expression is 0.

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