The six ratios in a right triangle

The six ratios in a right triangle

Build a geometric foundation for sine, cosine, tangent, cosecant, secant and cotangent. This precalculus practice set uses labelled right triangles and side data to connect each ratio with the chosen acute angle. It emphasizes identifying the opposite leg, adjacent leg and hypotenuse before calculating.

Questions include direct and reciprocal ratios, missing-side reconstruction with the Pythagorean theorem, scale comparisons and error analysis. Multiselect tasks require checking every proposed ratio. You will also examine whether a claimed triangle ratio agrees with the supplied side lengths, rather than accepting a memorized symbol pattern.

Prerequisites and practice

Review fractions, square roots, similar triangles and the Pythagorean theorem. Draw or annotate the right triangle, label the reference angle, and simplify exact values only after choosing the correct pair of sides. Worked explanations explain why a ratio stays unchanged under uniform scaling and why the hypotenuse is different from the adjacent leg. All angles in this initial geometric interpretation are acute. General-angle signs, unit-circle definitions and trigonometric equations belong to later practice.