= Pythagorean trigonometric identity
{c}
{title2=$\sin^2x+\cos^2x=1$}
{wiki}
For every real <angle> $\theta$,
$$
\boxed{\sin^2\theta+\cos^2\theta=1.}
$$
It is the equation of the <unit circle> written in terms of its <sine> and <cosine> coordinates. For an acute <angle>, dividing the <Pythagorean theorem> by the square of the <hypotenuse> gives the same relation in a <right triangle>. The circle interpretation explains why it remains true outside that first-quadrant triangle.
For example, if $\sin\theta=3/5$, the identity gives $\cos^2\theta=16/25$, so $\cos\theta$ is either $4/5$ or $-4/5$. The sign needs the <angle>'s location: it is positive in the first quadrant and negative in the second. Thus the identity determines the magnitude of the missing coordinate without determining the direction.
The relation is useful for eliminating one squared <trigonometric function>, checking that a coordinate pair has unit length, and checking that the columns of a <rotation matrix> are <unit vectors>. It also holds for complex arguments under the <complex exponential> definitions, although then the squares are algebraic squares rather than squared <moduli>.
Back to article page
