Question

PLEASE PLEASE PLEASE HELP!!! Where does the Pythagorean Identity sin2 Θ + cos2 Θ = 1 come from? How 
does it relate to right triangles?

Answer 1

All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Given that:

Sine function $f(\theta) = sin(\theta)$ being $sin(\theta)= v$ 
Cosine function $f(\theta) = cos(\theta)$ being $cos(\theta)= u$ 
$r = 1$ 

We will demonstrate the identity above. First of all, we need to square each equation, so:
 
$sin^{2}(\theta)= v^{2}$
$cos^{2}(\theta)=u^{2}$

Adding these two equations:

$sin^{2}(\theta)+cos^{2}(\theta)=v ^{2}+u^{2}$

But as shown in the figure, using Pythagorean theorem $v^{2}+u^{2}$ is always equal to 1, then:

$sin^{2}(\theta)+cos^{2}(\theta)=1$

The relation to right triangles is that:

The hypotenuse is always equal to 1
The opposite side is equal to $sin(\theta)$
The adjacent side is equal to $cos(\theta)$