7.3 Unit Circle

Unit Circle and Trigonometric Functions

The unit circle is a circle with radius 1, centered at the origin of the coordinate plane. Every point on it has coordinates $(\cos\theta,\, \sin\theta)$, which means the x-coordinate gives you cosine and the y-coordinate gives you sine. Once you understand how the unit circle works, you can evaluate trig functions for any angle without a calculator.

Sine and cosine for common angles

Three angles show up constantly in trig: 30°, 45°, and 60°. Their radian equivalents and trig values are worth memorizing outright.

A handy pattern: the sine values for 0°, 30°, 45°, 60°, 90° follow $\frac{\sqrt{0}}{2},\, \frac{\sqrt{1}}{2},\, \frac{\sqrt{2}}{2},\, \frac{\sqrt{3}}{2},\, \frac{\sqrt{4}}{2}$. The cosine values are the same sequence in reverse order.

Domain and range of trigonometric functions

• Domain of sine and cosine: all real numbers, $(-\infty, \infty)$. You can plug in any angle because rotation around the circle never has to stop.
• Range of sine and cosine: $[-1, 1]$. Since the unit circle has radius 1, the x- and y-coordinates can never be larger than 1 or smaller than -1.

This matters when you're solving equations. If someone asks you to solve $\sin\theta = 2$, you know immediately there's no solution, because 2 is outside the range.

Reference angles on the unit circle

A reference angle is the acute angle (between 0° and 90°) formed between the terminal side of your angle and the x-axis. It lets you connect any angle back to the familiar first-quadrant values you already memorized.

To find a reference angle:

1. Quadrant I (0° to 90°): The reference angle equals the angle itself.
2. Quadrant II (90° to 180°): Subtract the angle from 180° (or $\pi$).
3. Quadrant III (180° to 270°): Subtract 180° (or $\pi$) from the angle.
4. Quadrant IV (270° to 360°): Subtract the angle from 360° (or $2\pi$).

Examples:

• 120° is in Quadrant II → reference angle = $180° - 120° = 60°$
• $\frac{5\pi}{4}$ is in Quadrant III → reference angle = $\frac{5\pi}{4} - \pi = \frac{\pi}{4}$
• 315° is in Quadrant IV → reference angle = $360° - 315° = 45°$

Trigonometric functions in all quadrants

Once you have the reference angle, you evaluate sine or cosine of that reference angle, then attach the correct sign based on the quadrant. The mnemonic "All Students Take Calculus" helps you remember which functions are positive:

• Quadrant I (A): All trig functions positive
• Quadrant II (S): Sine positive, cosine negative
• Quadrant III (T): Tangent positive, sine and cosine both negative
• Quadrant IV (C): Cosine positive, sine negative

Step-by-step process:

1. Find the reference angle.
2. Evaluate sine or cosine of the reference angle (using your memorized values).
3. Determine which quadrant the original angle is in.
4. Apply the correct sign.

Example: Evaluate $\sin 120°$.

1. 120° is in Quadrant II → reference angle = $180° - 120° = 60°$

2. $\sin 60° = \frac{\sqrt{3}}{2}$

3. Sine is positive in Quadrant II.

4. Therefore, $\sin 120° = \frac{\sqrt{3}}{2}$

Example: Evaluate $\cos \frac{5\pi}{4}$.

1. $\frac{5\pi}{4}$ is in Quadrant III → reference angle = $\frac{5\pi}{4} - \pi = \frac{\pi}{4}$

2. $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

3. Cosine is negative in Quadrant III.

4. Therefore, $\cos \frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$

Unit Circle Properties and Motion

• The unit circle is centered at $(0, 0)$ with a radius of exactly 1.
• Any point on the circle satisfies the equation $x^2 + y^2 = 1$. This is actually the Pythagorean theorem at work, and it's why $\cos^2\theta + \sin^2\theta = 1$ is always true.
• Positive angles are measured counterclockwise from the positive x-axis; negative angles go clockwise.
• Trig functions are periodic: after a full rotation of $360°$ (or $2\pi$ radians), sine and cosine values repeat. That means $\sin\theta = \sin(\theta + 2\pi)$ and $\cos\theta = \cos(\theta + 2\pi)$ for any angle $\theta$.