4.2 The Unit Circle and Exact Values

Notes

On the unit circle, the point at angle $\theta$ is $(\cos\theta, \sin\theta)$ and $\displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta}$.

• First-quadrant values: $\displaystyle \sin \frac{\pi}{6} = \frac{1}{2}$, $\displaystyle \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\displaystyle \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$; cosine reverses this order.
• Signs by quadrant: sine is positive in I and II, cosine in I and IV, tangent in I and III.
• Use a reference angle (the acute angle to the x-axis) to find values in other quadrants.
• Example: $\displaystyle \cos\frac{5\pi}{4} = -\cos\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$.