Using Sum and Difference Formulas

Pre-Calculus · Axiom Academy

EXAMPLE Using Sum and Difference Formulas Find the exact value of a non-special angle by splitting it into special angles you already know. Find the exact value of . Since is not one of the special angles, write it as a difference of two angles whose sine and cosine you know, then apply the cosine difference formula. The exact values we will substitute once the angle is split. Nice work. You turned an "impossible" angle into exact radical form by decomposing it into angles you already know. Decompose: rewrite the awkward angle as a sum or difference of special angles — here . Mind the sign flip: . The cosine formula uses the opposite sign of the operation inside the angle. Same recipe, other functions: for use (negative, as expected in Quadrant II). And when you are given , (both Quadrant I), first recover , from , then . The whole toolkit is one idea: break an unknown angle into known pieces, pick the right formula, and substitute carefully — the quadrant decides every sign.

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