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 [tex]f(\theta) = sin(\theta)[/tex] being [tex]sin(\theta)= v[/tex]
Cosine function [tex]f(\theta) = cos(\theta)[/tex] being [tex]cos(\theta)= u[/tex]
[tex]r = 1[/tex]
We will demonstrate the identity above. First of all, we need to square each equation, so:
[tex]sin^{2}(\theta)= v^{2}[/tex]
[tex]cos^{2}(\theta)=u^{2}[/tex]
Adding these two equations:
[tex]sin^{2}(\theta)+cos^{2}(\theta)=v ^{2}+u^{2}[/tex]
But as shown in the figure, using Pythagorean theorem [tex]v^{2}+u^{2}[/tex] is always equal to 1, then:
[tex]sin^{2}(\theta)+cos^{2}(\theta)=1[/tex]
The relation to right triangles is that:
The hypotenuse is always equal to 1
The opposite side is equal to [tex]sin(\theta)[/tex]
The adjacent side is equal to [tex]cos(\theta)[/tex]
