Trig Identities You Need for the ACT

ACT Math · Axiom Academy

LESSON Trig Identities You Need for the ACT Which identities actually show up on the test — and the exact moves to turn a two-minute problem into a fifteen-second one. 1. Tier 1 — The Pythagorean Identity The single most important trig identity on the ACT comes straight from the Pythagorean theorem on the unit circle . A point at angle sits at , so its horizontal leg is , its vertical leg is , and the hypotenuse is the radius 1 . The legs squared always add to the radius squared — and the radius is 1 The identity has two rearranged forms that are just as useful — the ACT often gives you one function and wants the other: Example — using the Pythagorean identity If and is in Quadrant I, find . 2. Tier 1 — The Quotient Identity Tangent is just sine divided by cosine . Geometrically it's the slope of the line from the origin out to the point — rise over run, which is over . If you know two of the three functions, this gets you the third. tangent = rise ÷ run = the slope of the radius Example — using the quotient identity 3. Tier 2 — Reciprocal Identities Each of the three main functions has a reciprocal partner . A function and its reciprocal always multiply to 1 : when one grows, the other shrinks in exact proportion. Watch the see-saw — as climbs, drops, and their product stays pinned at 1 . Cosecant partners with sine. MEDIUM Secant partners with cosine. MEDIUM Cotangent partners with tangent. MEDIUM for every angle where it's defined. Example — a reciprocal identity

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