Interactive guide
sin(a+b) is not sin a + sin b. Not even close.
Both values, same number line.
O→B = cos b along the angle-a line; B→C = sin b, perpendicular.
cos b sin b radius 1, angle a+b
- sin a
- 0.7071
- sin b
- 0.7071
- guess
- 1.4142
- truth
- 1.0000
—
—
—
Quick gut check. Let a = b = 90°. Then sin a + sin b = 1 + 1 = 2, but sin(a+b) = sin(180°) = 0. Sine never exceeds 1, so "sin a + sin b" isn't just a little off, it's not even a value sine could ever produce. Slide either angle above down to 0° and watch the guess and the truth slide together, the only case where they agree.
The two identities, in full
Sine and cosine of a sum expand into a cross of the parts, not a sum of the parts:
sin(a+b) = sin a cos b + cos a sin b
cos(a+b) = cos a cos b − sin a sin b
Notice the pattern: sine's version crosses sin with cos and adds; cosine's version keeps cos with cos and sin with sin, and subtracts. Swap every + for a − and you get the angle difference formulas, sin(a−b) and cos(a−b), for free.
Reading the diagram on the right
-
1
O → B, length cos b
Along a rotated (angle-a) axis. Its height above the real x-axis is sin a cos b.
-
2
B → C, length sin b
Perpendicular to the rotated axis. Its extra height is cos a sin b; its horizontal shift is −sin a sin b.
-
3
C lands on the unit circle at a+b
Every time, because it's just two right triangles stacked, not a coincidence.
-
4
Stack heights for sin, x-positions for cos
The same two points give both formulas at once, one from C's height, one from its horizontal position.
So what is sin a + sin b, then, if not sin(a+b)? It's its own genuinely different identity: sin a + sin b = 2 sin((a+b)/2) cos((a−b)/2), a sum-to-product formula. It's a real, useful identity, it's just answering a different question than "what angle do you get by adding a and b."
The jargons
Two formulas, one shared construction. Everything below is a name for a piece of that picture.
A formula for sin or cos of a+b in terms of the sines and cosines of a and b separately.
Same formulas with every + replaced by − (and vice versa), for a−b instead of a+b.
A different family of formulas that turns sin a + sin b (an actual sum) into a product of sines and cosines of half-angle combinations.
A second x-axis, tilted to angle a, used to measure b relative to a instead of relative to the real x-axis.