Question

What is `3sin(theta/2)+cot(theta/2)` in terms of trigonometric functions of a unit `theta`?

Answer 1

`3sqrt((1-costheta)/2)+-sqrt((1+costheta)/(1-costheta))`,

with negative sign chosen, when `theta in Q_3 or Q_4`

Explanation:

If #theta in Q_1 or Q_2, theta/2 in Q_1, wherein

sine(theta/2) and cot(theta/2) are >=0#sine,

If #theta in Q_3 or Q_4, theta/2 in Q_2, wherein sin (theta/2) >=0,

but cot (theta/2) <=0#,

Use`cos 2A=2cos^2A-1=1-2sin^2A`.

The given expression is

`3sqrt((1-costheta)/2)+-sqrt((1+costheta)/(1-costheta))`,

with negative sign chosen, when `theta in Q_3 or Q_4`