The Sine Rule

The sine rule states that:

                                                

The triangle does not have to have a right angle in it for this rule to work.

So say I have this triangle:

 

We might want to find out what the angle opposite the side of length 20 is:

We can use the sine rule to solve for this angle, as follows:

                                              Let a = the side of length 50

                                       Let A = the angle opposite ‘a’ (100°)

                                              Let b = the side of length 20

                                            Let B = the angle opposite ‘b’

                                                    

Inverting both sides gives:

                                                    

Multiplying both sides by 20 gives:

                                              

On most calculators, there is an inverse button for sin, cos and tan. The inverse trigonometric symbol usually looks the name of the trigonometric ratio raised to the power negative one, something like ‘’.  The inverse button for ‘cos’ on a calculator looks something like   or . Since it’s above the button, you may have to press the ‘shift’ or ‘inverse function’ key first to use it. You use this function in situations such as this, when you know what sine of an angle is, and you want to find the angle itself.

To find out angle B, enter 0.3939 into the calculator, then press the inverse sin button.  You should get that the angle B is about 23.2 degrees.