If tan θ = 8/15, and θ is an acute angle, then find the value of √(1 - sin θ) / √(1 + sin θ).

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Accepted Answer

Given tan θ = 8/15 and θ is acute.

Step 1: Find sin θ and cos θ.

We know tan θ = sin θ / cos θ = 8/15.

Let sin θ = 8k and cos θ = 15k for some k.

Since sin² θ + cos² θ = 1,  
(8k)² + (15k)² = 1  
64k² + 225k² = 1  
289k² = 1  
k² = 1/289  
k = 1/17 (positive since θ is acute)

So, sin θ = 8/17 and cos θ = 15/17.

Step 2: Calculate the expression √(1 - sin θ) / √(1 + sin θ).

Substitute sin θ = 8/17:

Numerator: √(1 - 8/17) = √(9/17) = 3/√17  
Denominator: √(1 + 8/17) = √(25/17) = 5/√17

Expression = (3/√17) / (5/√17) = 3/5.

Therefore, the value is 3/5.

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