Solve cos^2 θ – sin θ – 1/ 4 = 0 (0° < θ < 360°)

Question

Solve cos² θ – sin θ – \(\frac{1}{4}\) = 0 (0° < θ < 360°)

Answer 1

cos² θ – sin θ - \(\frac{1}{4}\) = 0 

⇒ 1 – sin² θ – sin θ – \(\frac{1}{4}\)  = 0 

⇒ 4 sin² θ + 4 sin θ – 3 = 0 

⇒ (2 sin θ + 3) (2 sin θ – 1) = 0 

⇒ 2 sin θ + 3 = 0 or 2 sin θ – 1 = 0 

⇒ sin θ = - \(\frac{3}{2}\) or sin θ = \(\frac{1}{2}\) 

⇒ θ = 30°, 150°. 

Since |sin θ| = \(\frac{3}{2}\) is > 1, the value sin θ = - \(\frac{3}{2}\) is inadmissible. 

∴ θ = 30°, 150°.