Right Triangle Trigonometry

Pre-Calculus · Axiom Academy

LESSON Right Triangle Trigonometry SOH-CAH-TOA: how the three trig ratios are built from the sides of a right triangle — and why they match the unit circle. 1. Naming the Sides (relative to θ) Pick one acute angle and call it . Every side of the triangle now gets a name relative to that angle . Watch each side highlight and label itself in turn. The longest side, always across from the angle. It never depends on which acute angle you pick. The side directly across from — the one that does not touch the angle. The side next to that is not the hypotenuse — it forms the angle together with the hypotenuse. "Opposite" and "adjacent" are relative — pick the other acute angle and the two trade places. Only the hypotenuse stays put. 2. SOH-CAH-TOA: Building the Ratios Each trig function is just one side over another . The mnemonic SOH-CAH-TOA tells you which two. Watch each ratio pick out its two sides and assemble itself. S ine = O pposite / H ypotenuse C osine = A djacent / H ypotenuse T angent = O pposite / A djacent Dividing SOH by CAH cancels the hypotenuse: . 3. Work It Out: the 3-4-5 Triangle Now use actual numbers. For our angle , the opposite side is 3 , the adjacent is 4 , and the hypotenuse is 5 (and 3^2+4^2=5^2 , so it really is a right triangle). Substitute and reduce. 4. Why It Matches the Unit Circle Scale the triangle so the hypotenuse is exactly 1 . Watch it shrink onto a circle of radius 1 — the right-angle corner traces the point .

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