# Unit Circle Trigonometric Values

## Key Angles in Degrees and Radians:

- $0^\circ = 0$ radians
- $30^\circ = \frac{\pi}{6}$ radians
- $45^\circ = \frac{\pi}{4}$ radians
- $60^\circ = \frac{\pi}{3}$ radians
- $90^\circ = \frac{\pi}{2}$ radians
- $120^\circ = \frac{2\pi}{3}$ radians
- $135^\circ = \frac{3\pi}{4}$ radians
- $150^\circ = \frac{5\pi}{6}$ radians
- $180^\circ = \pi$ radians
- $210^\circ = \frac{7\pi}{6}$ radians
- $225^\circ = \frac{5\pi}{4}$ radians
- $240^\circ = \frac{4\pi}{3}$ radians
- $270^\circ = \frac{3\pi}{2}$ radians
- $300^\circ = \frac{5\pi}{3}$ radians
- $315^\circ = \frac{7\pi}{4}$ radians
- $330^\circ = \frac{11\pi}{6}$ radians
- $360^\circ = 2\pi$ radians

## Trigonometric Values at Key Angles:

| Angle (Degrees) | Angle (Radians)   | $ \cos(\theta) $      | $ \sin(\theta) $      | $ \tan(\theta) $      |
| --------------- | ----------------- | --------------------- | --------------------- | --------------------- |
| $0^\circ$       | $0$               | $1$                   | $0$                   | $0$                   |
| $30^\circ$      | $\frac{\pi}{6}$   | $\frac{\sqrt{3}}{2}$  | $\frac{1}{2}$         | $\frac{1}{\sqrt{3}}$  |
| $45^\circ$      | $\frac{\pi}{4}$   | $\frac{\sqrt{2}}{2}$  | $\frac{\sqrt{2}}{2}$  | $1$                   |
| $60^\circ$      | $\frac{\pi}{3}$   | $\frac{1}{2}$         | $\frac{\sqrt{3}}{2}$  | $\sqrt{3}$            |
| $90^\circ$      | $\frac{\pi}{2}$   | $0$                   | $1$                   | $\text{undefined}$    |
| $120^\circ$     | $\frac{2\pi}{3}$  | $-\frac{1}{2}$        | $\frac{\sqrt{3}}{2}$  | $-\sqrt{3}$           |
| $135^\circ$     | $\frac{3\pi}{4}$  | $-\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$  | $-1$                  |
| $150^\circ$     | $\frac{5\pi}{6}$  | $-\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$         | $-\frac{1}{\sqrt{3}}$ |
| $180^\circ$     | $\pi$             | $-1$                  | $0$                   | $0$                   |
| $210^\circ$     | $\frac{7\pi}{6}$  | $-\frac{\sqrt{3}}{2}$ | $-\frac{1}{2}$        | $\frac{1}{\sqrt{3}}$  |
| $225^\circ$     | $\frac{5\pi}{4}$  | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $1$                   |
| $240^\circ$     | $\frac{4\pi}{3}$  | $-\frac{1}{2}$        | $-\frac{\sqrt{3}}{2}$ | $\sqrt{3}$            |
| $270^\circ$     | $\frac{3\pi}{2}$  | $0$                   | $-1$                  | $\text{undefined}$    |
| $300^\circ$     | $\frac{5\pi}{3}$  | $\frac{1}{2}$         | $-\frac{\sqrt{3}}{2}$ | $-\sqrt{3}$           |
| $315^\circ$     | $\frac{7\pi}{4}$  | $\frac{\sqrt{2}}{2}$  | $-\frac{\sqrt{2}}{2}$ | $-1$                  |
| $330^\circ$     | $\frac{11\pi}{6}$ | $\frac{\sqrt{3}}{2}$  | $-\frac{1}{2}$        | $\frac{1}{\sqrt{3}}$  |
| $360^\circ$     | $2\pi$            | $1$                   | $0$                   | $0$                   |

## Patterns to Notice:

1. **Sine and Cosine Symmetry:**
   - The values of sine and cosine are symmetric in different quadrants.
   - For example, sine values are positive in the first and second quadrants,
     negative in the third and fourth quadrants.
   - Cosine values are positive in the first and fourth quadrants, negative in
     the second and third quadrants.

2. **Tangent Behavior:**
   - Tangent values are undefined when cosine equals zero (at $90^\circ$ and
     $270^\circ$).
   - Tangent has a period of $180^\circ$, so the values for angles like
     $45^\circ$ and $225^\circ$ are the same.