Apply the sine, cosine, and tangent ratios (and their inverses) to find a missing side or angle of a right triangle.
Every right triangle has, besides its 90-degree angle, two acute angles, and the RATIOS between its sides depend only on those angle measures, not on the triangle's size (similar right triangles share every ratio, by AA Similarity). The three trigonometric ratios name these relationships relative to a chosen acute angle theta: sine (sin) is opposite over hypotenuse, cosine (cos) is adjacent over hypotenuse, and tangent (tan) is opposite over adjacent. The memory phrase SOH-CAH-TOA packs all three definitions into one word.
Trigonometry runs in two directions. If you know an acute angle and one side, multiply or divide by the right ratio to find a missing side. If you know two sides but not the angle, use the INVERSE trig functions (sin^-1, cos^-1, tan^-1 -- a calculator's "2nd" versions of the same buttons) to recover the angle from the ratio. Before choosing sin, cos, or tan for any problem, always label the triangle first: find the right angle, mark the given acute angle theta, and only then label which sides are opposite, adjacent, and the hypotenuse relative to THAT angle.
Mastery looks like: They correctly label a right triangle relative to a given angle and choose the correct trig ratio (or its inverse) to find a missing side or angle.
Common stumbles: Mixing up which leg is opposite versus adjacent when theta is at a different vertex than expected; forgetting to use the INVERSE trig function when solving for an angle instead of a side.
See it visually
Interactive, spoken-aloud tools that show this idea in motion.