Dealing with half angles

Question

How do you determine whether the sign of a trigonometric function (sine, cosine, tangent) is positive or negative when dealing with half angles? Why do the half-angle identities include positive and negative options but the other identities don't seem to have this option built in?

Answer 1

Depending on the angle, it is possible that the original angle is in one quadrant, but HALF of it is

in another quadrant, which then changes the sign of the trig function.

To determine the sign of the trig function, use the mnemonic device: ADD SUGAR TO COFFEE or ALL STUDENTS TAKE CALCULUS

A All trig functions positive in quadrant #1

S Sine function is positive in quadrant #2

T Tangent function is positive in quadrant #3

C Cosine function is positive in quadrant #4