Question

What is `2sin(theta/4)-cos(theta/2)` in terms of trigonometric functions of a unit `theta`?

Answer 1

`2sin(theta/4)-cos(theta/2)=sqrt(2-sqrt(1+costheta))-sqrt((1+costheta)/2)`

Explanation:

We can use the identities `cosA=sqrt((1+cos2A)/2)` and `sinA=sqrt((1-cos2A)/2)`

Hence, `2sin(theta/4)-cos(theta/2)`

= `2sqrt((1-cos(theta/2))/2)-sqrt((1+costheta)/2)`

= `2sqrt((1-sqrt(1+costheta)/2)/2)-sqrt((1+costheta)/2)`

= `2sqrt((2-sqrt(1+costheta))/4)-sqrt((1+costheta)/2)`

= `sqrt(2-sqrt(1+costheta))-sqrt((1+costheta)/2)`