Question

If \(2\cot \theta = 3\), then \(\frac{{\sqrt {13} \cos \theta - 3\tan \theta }}{{3\tan \theta + \sqrt {13} \sin \theta }}\) is:
1. 3/4
2. 1/4
3. 2/3
4. 1/5

Answer 1

Correct Answer - Option 2 : 1/4

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 above values in equation \(\frac{{\sqrt {13} \cos \theta - 3\tan \theta }}{{3\tan \theta + \sqrt {13} \sin \theta }}\)

⇒ \(\frac{{\sqrt {13} (3/\sqrt13) - 3(2/3) }}{{3(2/3) + \sqrt {13} (2/ \sqrt {13}) }}\)

⇒ (3 - 2)/(2 + 2) = 1/4 

∴ The value of \(\frac{{\sqrt {13} \cos \theta - 3\tan \theta }}{{3\tan \theta + \sqrt {13} \sin \theta }}\)is 1/4.