3.3 The Unit Circle

Notes

The unit circle has radius 1 and center at the origin, so for a point $(x, y)$ at angle $\theta$: $\cos\theta = x$ and $\sin\theta = y$.

• Quadrant I points: $0^\circ \to (1, 0)$; $\displaystyle 30^\circ \to \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$; $\displaystyle 45^\circ \to \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$; $\displaystyle 60^\circ \to \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$; $90^\circ \to (0, 1)$.
• Other quadrants use the same numbers with signs changed.
• $\displaystyle \tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta}$.
• Example: $\displaystyle \frac{2\pi}{3}$ (that is $120^\circ$) is the point $\displaystyle \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.