Question

Sin theta = 1/4 find cos (theta/2)

Answer 1

Use the pythagorean identity to find cos Θ.
sin² Θ + cos² Θ = 1
(¼)² + cos² Θ = 1
cos Θ = ±(√15)/4

Now use the half angle identity

`cos ((\theta)/(2))=\pm \sqrt{(1+cos\theta)/(2)`

This is why you need to know the quadrant. If Θ is in Q1, then cos Θ is (√15)/4. If Θ is in Q2, then cos Θ = -(√15)/4.

Also, since Θ is in either Q1 or Q2, Θ/2 must be in Q1, and cos (Θ/2) is positive

For Θ in Q1:
`cos((\theta)/(2))= \sqrt{(1+( √(15) )/(4))/(2)}`

For Θ in Q2:
`cos((\theta)/(2))= \sqrt{(1+(-( √(15) )/(4)))/(2)} \\ =cos((\theta)/(2))= \sqrt{(1-( √(15) )/(4))/(2)}`