Sine and Cosine Rule

In this post, I will discuss the derivations for the sine and cosine rule. Unlike right-angled trigonometry, these two rules are applicable to all triangles, regardless of lengths or angles.

From the Pythagorean Theorem:

And we know that x and y are:

And y:

remember that c=y+adj; adj is short for the unlabelled side that is adjacent to angle 

We can now substitute these formulae into the formula for a:

Expanding the brackets:

Factorising the b squared terms:

From the Pythagorean Identity:

Which is the cosine rule. It doesn't matter which side is a,b or c providing the angle A is opposite side a, the angle B opposite side b etcetera. The formula states that the side opposite to an angle can be calculated from the angle itself and the two adjacent sides.

For the sine rule, we'll use the same triangle changed a bit:

From the triangle we can see (from line y):

Which implies:

And from line x:

So:

Combining the two equations, we have:

Taking the reciprocal of each term also gives:

I hope you have understood these derivations of the sine and cosine rules. Don't be afraid to use the links if you didn't, and keep referring back to the triangle if need be. With thanks to desmos.-naBla5040