Angle Sum Formulas

Algebra 2 · Axiom Academy

LESSON Angle Sum and Difference Formulas Stack two angles on the unit circle and read off the sum, difference, and double-angle identities. Place angle on the unit circle, then stack angle on top so the terminal point P sits at angle . One dropped perpendicular splits P 's coordinates into two products each — giving both sum formulas at once. Dividing by gives the tangent sum formula. A difference is a sum with a negative angle: . On the circle that reflects the point across the -ray. Because but , every flips sign — and nothing else does. Now set . The stacked angle collapses into , and the mixed products fold together into the double-angle identities. 4. Exact Values, such as cos 75° The payoff: split an unfamiliar angle into two you already know. Since , the sum formula turns into products of exact values — no calculator required. Any angle that is a sum or difference of , , and — such as , , or — now has an exact value built from radicals alone. Every identity here came from one idea: stack two angles on the unit circle and read the coordinates of the point.

This is the written version of the interactive lesson above. See the full Algebra 2 course.