Deriving special-triangle values

Deriving special-triangle values

Derive exact trigonometric values from the geometry of 30-60-90 and 45-45-90 triangles. This precalculus practice set begins with an equilateral-triangle altitude and an isosceles right triangle, using the Pythagorean theorem to establish the side relationships behind familiar special-angle values.

Practice exact sine, tangent and reciprocal ratios, scale triangle sides, and recover side lengths when the hypotenuse or longer leg is known. Multiselect questions connect several consequences of one geometric model. An area application and a rationalization error task require transferring the relationships beyond a memorized value table.

Study method

Review the six right-triangle ratios, square roots and equivalent fractions. Keep radical expressions exact unless an approximation is explicitly requested. Draw the relevant triangle, label the angle opposite each side, and check that the hypotenuse is longest. The explanations retain the derivations alongside the values so you can reconstruct a forgotten ratio. All angles here are acute; quadrant signs, general-angle identities and trigonometric equations are outside this set.