How is trigonometry used on non-right angled triangles?

To do this, there are two rules, the Sine Rule and The Cosine Rule. 

The sine rule is a/Sin A   = b/Sin B = c/Sin C.  (the lower and uppercase are very important. The rule also stands if you write the entire thing the other way up.

The sine rule is used when you either have the vlaue of two angles and one side, or  two sides and one angle opposite one of those sides of a triangle. 

Label the triangle clockwise starting with the angles. Use uppercase (A) to label the angles. Next, label the sides opposite each angle with its respective lowercase letter (a)

Then simply input the values you have into the correct places of the formula. 

If you cannot use the sine rule, use The Cosine Rule:

a² = b² + c² - 2bc cos A

b² = a² + c² - 2ac cos B

c² = a² + b² - 2ab cos C

You must work out which of these to use depending on what values you are given in your question. 

To use the cosine rule you either need the length of all 3 sides, or the length of 2 sides and angle.  Label your triangle in the same way as you would using the Sine rule.  Then, input the numbers into the formula you have chosen. 

If you are left with Cos/ Sin/ Tan x, remember to use the inverse to get the answer. E.g. if Cos A = 0.764, A is found by pressing the Cos-1 sign on your caluclated to find the inverse of Cos 0.764. I.e 40.2.