Right triangle trigonometry

In a right triangle, fixing one acute angle fixes the shape of the triangle, so the ratios between its sides depend only on that angle. Those ratios are the trigonometric functions. Relative to a chosen acute angle θ, the side across from it is the opposite, the side touching it (other than the hypotenuse) is the adjacent, and the side across from the right angle is always the hypotenuse.

The three primary ratios are captured by the mnemonic SOHCAHTOA: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent. Which side is opposite versus adjacent flips when you switch reference angles — the most common source of error. To find a missing side, set up the ratio that uses the side you know and the side you want; to find a missing angle, use an inverse function such as θ = tan⁻¹(opposite / adjacent). The Pythagorean theorem, a² + b² = c², handles cases where two sides are known and no angle is needed.

Two special right triangles come up constantly and need no calculator. A 45-45-90 triangle has sides in the ratio 1 : 1 : √2, and a 30-60-90 triangle has sides in the ratio 1 : √3 : 2, with the shortest side opposite the 30° angle. These give the exact values sin 30° = 1/2, sin 45° = √2/2, and sin 60° = √3/2. The cofunction relationship sin θ = cos(90° − θ) is commonly tested, and extending these ratios beyond 90° is what the unit circle does.

Angle-of-elevation, ladder, ramp, and surveying problems all reduce to a right triangle. In physics, resolving a vector into components is right triangle trigonometry with the vector as the hypotenuse: the horizontal component is magnitude · cos θ and the vertical component is magnitude · sin θ. The SAT and CLT both test it in their geometry sections, including the special triangles and cofunction identities, and AP Physics 1 relies on it throughout vector components analysis and inclined plane problems.