How do you find the angle measurements of a triangle if you have the side lengths?

Answer 1

The way to find the angle measures of a triangle if you have the side lengths is to use inverse trigonometry.

If a triangle is a right triangle (meaning it has one right angle or 90 degree angle) then you can use Right Triangle Trigonometry.

There are three trigonometric (or trig) functions that we can use: Sin (pronounced Sign, short for sine), Cos (short for cosine), and Tan (short for tangent). These are all names of functions. These functions relate an angle measure of a right triangle to the ratio of two particular sides.

Generally SOH CAH TOA is the mnemonic device people use to remember the trig ratios. Sin(x) = Opposite/Hypotenuse Cos(x) = Adjacent/Hypotenuse Tan(x) = Opposite/Adjacent. In each of these relationships, x is the value of one of the acute angles of the triangle.

We want to however, find the angle measure rather than the ratio of the sides. So to do this, we isolate the value of the angle by taking the inverse function of both sides, resulting in the following equations where x is the value of the acute angle:

x = sin-1(opposite/hypotenuse) or x = cos-1(adjacent/hypotenuse) or x = tan-1(opposite/adjacent). The negative one that you see here is just denoting that you need to use the inverse of the function. Most calculators that have the trig functions available, also have the inverses available as well.

As a quick example, take the following right triangle ABC:

A

|\

|.\

|..\

|...\

|....\

|.....\

C----B

If AB=5, BC=3, and AC=4 and we know that C is a right angle, we can find angle B by doing the following calculation on the calculator:

the measure of angle B=tan-1(4/3)

The calculator would yield an answer of roughly 53.1 degrees.

If the triangle is not a right triangle, then you either have to use the law of cosine or the law of sin, both of which are very well explained on wikipedia. They are simply ways to relate side lengths to angle measures using the trig functions.