Question

If secθ - cosθ = 1/2, find the value of sin²θ when θ lies in first quadrant.

1. (√15 – 6)/8
2. (√17 – 3)/8
3. (√17 – 1)/8
4. (√14 – 7)/8

Answer 1

Correct Answer - Option 3 : (√17 – 1)/8

Given:

secθ - cosθ = 1/2

Concepts used:

sin²θ + cos²θ = 1

secθ = 1/cosθ

Calculation:

secθ - cosθ = 1/2

⇒ (1/cos²θ) – cosθ = 1/2

⇒ 1 – cos²θ = cosθ/2

⇒ 2 – 2cos²θ – cosθ = 0 

⇒ 2cos²θ + cosθ – 2 = 0

θ lies in 1st quadrant

⇒ cosθ is positive in first quadrant.

⇒ cosθ = (√17 – 1)/4

sin²θ =  1 – cos²θ

⇒ sin²θ = 1 – {(√17  - 1)/4}²

⇒ sin²θ = 1 – {(9 - √17)/8}

⇒ sin²θ = (√17 – 1)/8

∴ Value of sin²θ is (√17 – 1)/8.