Basic Trigonometric Ratios (page 1 of 2)

Right triangles are nice and neat, with their side lengths obeying the Pythagorean Theorem. Any two right triangles with the same two non-right angles are "similar", in the technical sense that their corresponding sides are in proportion. For instance, the following two triangles (not drawn to scale) have all the same angles, so they are similar, and the corresponding pairs of their sides are in proportion:

two similar right triangles, with sides 6, 8, 10 and 3, 4, 5

ratios of corresponding sides: 10/5 = 8/4 = 6/3 = 2 ratio of hypotenuse to base: 10/8 = 5/4 = 1.25 ratio of hypotenuse to height: 10/6 = 5/3 = 1.666... ratio of height to base: 6/8 = 3/4 = 0.75

Around the fourth or fifth century AD, somebody very clever living in or around India noticed these consistency of the proportionalities of right triangles with the same sized base angles, and started working on tables of ratios corresponding to those non-right angles. There would be one set of ratios for the one-degree angle in a 1-89-90 triangle, another set of ratios for the two-degree angle in a 2-88-90 triangle, and so forth. These ratios are called the "trigonometric" ratios for a right triangle.

Given a right triangle with a non-right angle designated as θ ("THAY-tuh"), we can label the hypotenuse (always the side opposite the right angle) and then label the other two sides "with respect to θ" (that is, in relation to the non-right angle θ that we're working with).

The side opposite the angle θ is the "opposite" side, and the other side, being "next" to the angle (but not being the