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:
ratios of corresponding sides: 10/5 = 8/4 = 6/3 = 2
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.
Let's return to the first labelling:
There are six ways to form ratios of the three sides of this triangle. I'll shorten the names from "hypotenuse", "adjacent", and "opposite" to "hyp", "adj", and "opp":
Each of these ratios has a name:
...and each of these names has an abbreviated notation, specifying the angle you're working with:
The ratios in the left-hand table are the "regular" trig ratios; the ones in the right-hand table are their reciprocals (that is, the inverted fractions). To remember the ratios for the regular trig functions, many students use the mnemonic SOH CAH TOA, pronounced "SOH-kuh-TOH-uh" (as though it's all one word). This mnemonic stands for:
When two ratios have the same name other than the "co-" at the start of one of them, the pair are called "co-functions". (Note: The ratios didn't get their current names until the 12th century or so, as a result of Europeans making some mistakes when they translated Arabic texts.)
Original URL: http://www.purplemath.com/modules/basirati.htm Copyright 2009 Elizabeth Stapel; All Rights Reserved.
Copyright 2009 Elizabeth Stapel; All Rights Reserved.