How to solve a triangle with three sides

How to solve a triangle given three sides. This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how to find the unknown angles of a triangle when given the threeside lengths.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.Theory:

This triangle has internal angles 'A', 'B' and 'C', and sides of length 'a', 'b' and 'c':

If three of these six measurements are known, then it may be possible to find the otherthree.

This is called 'solving' the triangle, and this topic will show you how to solve triangles for the three unknown angles when the three side lengths 'a', 'b' and 'c' are known.



These arethe formulas used to solve triangles:



1. The sum of the internal angles equals 180º ...

A + B + C = 180º



2. The 'sine rule' ...

3. The 'cosine rule' ...

a² = b² + c² -2bc cosA

or

b² = a² + c² - 2ac cosB

or

c² = b² + a² - 2ba cosC

We will now use an example to show how these rules are applied to solve a triangle when the threeside lengths are given.



Example: Solve this triangle for the unknown internal angles:

When no angles are known, the cosine rule is the only option.

Step 1: Begin by using the cosine ruleto find the largest angle.

NOTE: We find the largest angle first, because there can only be one angle in a triangle that is obtuse (greater than 90°). If a triangle has an obtuse angle, then thiswill be it. The reason for finding it first is that in the next step we will use the sine rule to find the second angle. The inverse sin operation that we will use can only give us acute angles (lessthan 90°), so we avoid a possible wrong answer by first eliminating the only possibility of an obtuse angle.

The largest angle is always opposite to the largest side, so this is angle 'C' in this...