5.2 The Unit Circle

Notes

The unit circle has radius 1 and center at the origin. The point at angle $\theta$ is $(\cos\theta, \sin\theta)$.

• Key values: $\displaystyle \sin\frac{\pi}{6} = \frac{1}{2}$, $\displaystyle \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$; $\displaystyle \sin\frac{\pi}{4} = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$; $\displaystyle \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$, $\displaystyle \cos\frac{\pi}{3} = \frac{1}{2}$.
• Use the reference angle and the quadrant signs. Sine is positive in quadrants I and II. Cosine is positive in I and IV.
• Axes: $\cos\pi = -1$, $\displaystyle \sin\frac{3\pi}{2} = -1$, $\cos 0 = 1$.
• Since the point is on the circle, $\cos^2\theta + \sin^2\theta = 1$.