how do you find the +/- of this equation cos(a/2)=+/- √[(1 + CosA) / 2]?

Question

like when you get the answer, how do you figure out if the answer is +/- ?

Answer 1

cos²(a/2)=(1+cos(a))/2,

2cos²(a/2)=1+cos(a),

cos(a)=2cos²(a/2)-1, which is an identity.

So given cos(a) you can find cos(a/2). The plus/minus simply means that both the positive and negative roots give you cos(a/2).

Example: a=60°, then cos(a)=½, cos(a/2)=±√¾=±½√3. This makes a/2=30°, the positive sign in this case because we already knew a. But if you're given cos(a) rather than a itself, there's an ambiguity, because a is not unique when you know cos(a). If cos(a)=½, a could be 60° or 300°, and other values, too (420°, 660°, etc.). So, for example, a/2 could be 30° (which has a positive cosine) or 150° (which has a negative cosine). To decide whether you apply plus or minus you need more information, otherwise you can assume that both plus and minus are valid.