the angle-sum and double-angle identities

You can read sin 30 degrees and sin 45 degrees straight off the unit circle, but what about sin 75 degrees? You cannot just add the two sines — sine does not split over addition. The angle-sum identities are the correct rules for combining angles, and they are the workhorses behind almost every trigonometric manipulation.

The two fundamental formulas are sin(A + B) = sin A cos B + cos A sin B and cos(A + B) = cos A cos B - sin A sin B (with the signs flipped for A - B). Notice cosine's formula mixes a minus sign in. Setting B = A turns each into a double-angle identity: sin 2A = 2 sin A cos A, and cos 2A = cos^2 A - sin^2 A, which the Pythagorean identity also rewrites as 1 - 2 sin^2 A or 2 cos^2 A - 1. Solving those last two for sin^2 A and cos^2 A and halving the angle gives the half-angle identities, such as sin^2(A/2) = (1 - cos A) / 2. So sin 75 degrees = sin(45 + 30) = sin 45 cos 30 + cos 45 sin 30 = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = (sqrt(6) + sqrt(2)) / 4.

These identities let you evaluate new angles exactly, collapse complicated expressions, and reverse the process to factor sums into products. They underlie the addition of waves in physics and signal processing — two sinusoids of the same frequency add to a single shifted sinusoid precisely because of these formulas. The cardinal sin is to assume sin(A + B) = sin A + sin B; it is false, and these formulas exist precisely to replace that wrong guess.