Skill: Use the unit circle to evaluate sine and cosine

On the unit circle, an angle $\theta$ in standard position ends at a point

$$(\cos\theta,\sin\theta).$$

So:

• $\cos\theta$ is the $x$-coordinate.
• $\sin\theta$ is the $y$-coordinate.

To evaluate a familiar angle, identify its quadrant and use its reference angle.

Example: Evaluate $\cos\left(\frac{5\pi}{6}\right)$ and $\sin\left(\frac{5\pi}{6}\right)$.

Step 1: Identify the quadrant.

The angle $\frac{5\pi}{6}$ is between $\frac{\pi}{2}$ and $\pi$, so it lies in Quadrant II.

In Quadrant II:

• cosine is negative,
• sine is positive.

Step 2: Find the reference angle.

The reference angle is the distance from $\frac{5\pi}{6}$ to $\pi$:

$$\pi-\frac{5\pi}{6}=\frac{\pi}{6}.$$

The unit-circle point for $\frac{\pi}{6}$ is

$$\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right).$$

Thus, the reference values are

$$\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}, \qquad \sin\left(\frac{\pi}{6}\right)=\frac{1}{2}.$$

Step 3: Apply the Quadrant II signs.

Cosine must be negative, while sine remains positive:

$$\boxed{\cos\left(\frac{5\pi}{6}\right)=-\frac{\sqrt{3}}{2}}$$

and

$$\boxed{\sin\left(\frac{5\pi}{6}\right)=\frac{1}{2}}.$$

The terminal point on the unit circle is therefore

$$\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right),$$

where the first coordinate gives cosine and the second gives sine.