Worked example

Given cos x = −3/5 with 180° < x < 270°, find sin(x/2) and cos(x/2)

Answer

sin(x/2) = √(4/5) = 2/√5 and cos(x/2) = −1/√5

180° < x < 270° → 90° < x/2 < 135°
Halve the interval first — the sign of the half angle is decided there.
sin(x/2) = +√((1 − cos x)/2)
x/2 is in Q2, where sine is positive. · Half-angle identity for sine
= √((1 + 3/5)/2) = √(4/5) = 2/√5
Substitute cos x = −3/5.
cos(x/2) = −√((1 + cos x)/2)
Q2 cosine is negative — this is where the ± must be resolved. · Half-angle identity for cosine
= −√((2/5)/2) = −1/√5
Substitute and simplify.
Check: (2/√5)² + (1/√5)² = 4/5 + 1/5 = 1
Pythagorean identity holds ✓. · Pythagorean identity

Also the reason our identity pages keep the `conditions` field: the '+' root is only correct on a stated interval.

sin(x/2) = ±√((1 − cos x)/2)

cos(x/2) = ±√((1 + cos x)/2)

Half-angle identity for sine — A half angle appears, or an integral contains √(1 − cos x). Always decide the sign from the quadrant of x/2 BEFORE writing the root — that decision is the whole problem.
Half-angle identity for cosine — Half-angle cosine, or √(1 + cos x) under an integral. Same sign discipline as the sine version: the quadrant of x/2 decides.

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.