Sum and Difference Formulas

Pre-Calculus · Axiom Academy

LESSON Sum and Difference Formulas Find the exact value of trig functions at angles like 15° and 75° by writing them as sums or differences of the angles you already know. 1. Break a New Angle into Known Ones We can't read straight off the unit circle — isn't one of the special angles. But , and we do know both and . Watch the terminal ray sweep through a known , then stack another known on top to land exactly on . A difference works just as well For sine, every term mixes one sine and one cosine — and — and the sign in the middle is the same as the sign between the angles. In the animation, flip the angle sign and watch the middle operator follow it. Cosine pairs the same functions together — and — but its middle sign is the opposite of the sign between the angles. This is the one rule students miss most, so the animation makes it the whole point: as the angle sign toggles , the formula's middle sign toggles the other way. 4. Tangent — A Fraction with Opposite Signs Tangent is a single fraction. The top takes the same sign as the angles, ; the bottom is 1 minus or plus with the opposite sign. The animation lights the top and bottom signs together so you see them move in opposite directions. 5. Worked Example — Find the Exact Value of Put it together. Write and apply the cosine difference formula — so its middle sign flips to + . The animation substitutes the four known unit-circle values one at a time and simplifies to a single exact answer.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.