The Pythagorean Identity
Valid for every real number $\theta$.
This single equation is the most important identity in trigonometry. It connects the sine and cosine of any angle through a beautifully simple relationship: their squares always add up to 1. We will derive it two ways — first from the Pythagorean theorem in a right triangle (which works for acute angles), and then from the equation of the unit circle (which works for all real numbers).
Consider a right triangle whose hypotenuse has length 1. Place the triangle on the unit circle with one acute angle $\theta$ at the origin.
- The adjacent side (along the $x$-axis) has length $\cos\theta$.
- The opposite side (along the $y$-axis) has length $\sin\theta$.
- The hypotenuse equals the radius of the unit circle, which is $1$.
The Pythagorean theorem says $a^{2}+b^{2}=c^{2}$. Substituting $a=\sin\theta$, $b=\cos\theta$, $c=1$:
$$\sin^{2}\theta + \cos^{2}\theta = 1^{2} = 1$$The unit circle is the set of all points $(x,y)$ satisfying
$$x^{2}+y^{2}=1.$$By definition, for any real number $\theta$ the point on the unit circle at angle $\theta$ from the positive $x$-axis has coordinates
$$x = \cos\theta, \qquad y = \sin\theta.$$Because this point lies on the circle, it must satisfy the circle's equation. Substituting: