Sine and Cosine on the Unit Circle

The coordinate definition

Place an angle $\theta$ in standard position. Where its terminal ray meets the unit circle, the point is by definition $(\cos\theta,\ \sin\theta)$: cosine is the $x$-coordinate, sine is the $y$-coordinate. Everything else in the unit follows from this one sentence.

On a circle of radius $r$, the point becomes $(r\cos\theta,\ r\sin\theta)$; conversely, if the terminal ray passes through $(x,y)$ with $r=\sqrt{x^2+y^2}$, then $\cos\theta=\frac{x}{r}$ and $\sin\theta=\frac{y}{r}$.

Special values and signs

The first-quadrant values for $\theta=0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}$ generate all others through reference angles: find the acute angle to the $x$-axis, take the first-quadrant value, then attach the sign of the quadrant. Signs follow directly from coordinates: sine positive where $y>0$ (quadrants I, II), cosine positive where $x>0$ (quadrants I, IV).

Because the coordinates live on the unit circle, $\sin^2\theta+\cos^2\theta=1$ is just the Pythagorean theorem — and it converts one ratio into the other, up to a sign that the quadrant decides.