Pythagoras and Trigonometry

This is a short post.

There are many relations between Pythagoras and Trigonometry, but while writing the previous article, something occurred to me that I never thought of before.

I'm referring to this threesome of identities:

I know they're called the Pythagorean Identities, but I hadn't thought of them that way before. So I proceeded to draw triangles.

A triangle in a unit circle is particularly useful because the hypotenuse is of length 1 unit. This means that the lengths of the other sides can be represented with the ratios, without any multipliers attached to them.

If the 1 were replaced with some other number, like x, then the sides will be x sin θ and x cos θ, following the formulae of vector resolution.

Applying Pythagoras' Theorem to this right angled triangle, we get this:

The other identities require drawing the circle.

Applying Pythagoras' Theorem to this right angled triangle, we get this:

And finally:

So the last identity:

Just something interesting I wanted to share :)