The Unit Circle

What is the Unit Circle?

The unit circle is a circle with:

• Center at the origin $(0, 0)$
• Radius of exactly 1

Its equation is: $x^2 + y^2 = 1$

Why the Unit Circle Matters

The unit circle allows us to define trigonometric functions for all angles, not just acute angles in right triangles.

For any angle $\theta$ in standard position (starting from the positive x-axis):

• $\cos(\theta)$ = x-coordinate of the point on the unit circle
• $\sin(\theta)$ = y-coordinate of the point on the unit circle
• $\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}$

Special Angles on the Unit Circle

The Key Angles to Memorize

You should know these angles in both degrees and radians:

Pattern Recognition

For 30° and 60° angles:

• Think of the ratios: $\frac{1}{2}$, $\frac{\sqrt{3}}{2}$
• At $30°$: sine is small ($\frac{1}{2}$), cosine is large ($\frac{\sqrt{3}}{2}$)
• At $60°$: sine is large ($\frac{\sqrt{3}}{2}$), cosine is small ($\frac{1}{2}$)

For 45° angles:

• Everything is $\frac{\sqrt{2}}{2}$ (except tangent = 1)
• This makes sense: at $45°$, x and y coordinates are equal

All Four Quadrants

Once you know the first quadrant angles, you can find any angle using reference angles and the CAST rule!

Common angles in all quadrants (in radians):

• Quadrant I ($0$ to $\frac{\pi}{2}$): $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$
• Quadrant II ($\frac{\pi}{2}$ to $\pi$): $\frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}, \pi$
• Quadrant III ($\pi$ to $\frac{3\pi}{2}$): $\frac{7\pi}{6}, \frac{5\pi}{4}, \frac{4\pi}{3}, \frac{3\pi}{2}$
• Quadrant IV ($\frac{3\pi}{2}$ to $2\pi$): $\frac{5\pi}{3}, \frac{7\pi}{4}, \frac{11\pi}{6}, 2\pi$

Reference Angles

A reference angle is the acute angle formed between the terminal side of the angle and the x-axis.

Reference angles help you find trig values for angles in any quadrant!

Finding Reference Angles

Let $\theta$ be your angle. The reference angle $\theta'$ is:

• Quadrant I: $\theta' = \theta$
• Quadrant II: $\theta' = \pi - \theta$ (or $180° - \theta$)
• Quadrant III: $\theta' = \theta - \pi$ (or $\theta - 180°$)
• Quadrant IV: $\theta' = 2\pi - \theta$ (or $360° - \theta$)

Examples

Example 1: Find the reference angle for $\frac{5\pi}{6}$

This is in Quadrant II, so: $\theta' = \pi - \frac{5\pi}{6} = \frac{6\pi}{6} - \frac{5\pi}{6} = \frac{\pi}{6}$

Example 2: Find the reference angle for $240°$

This is in Quadrant III, so: $\theta' = 240° - 180° = 60°$

Example 3: Find $\cos(\frac{5\pi}{4})$

• $\frac{5\pi}{4}$ is in Quadrant III
• Reference angle: $\frac{5\pi}{4} - \pi = \frac{\pi}{4}$
• We know $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$
• In Quadrant III, cosine is negative
• Therefore: $\cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}$

CAST Rule (Signs by Quadrant)

CAST tells you which trig functions are positive in each quadrant:

• Quadrant I: All (sine, cosine, tangent all positive)
• Quadrant II: Sine positive only
• Quadrant III: Tangent positive only
• Quadrant IV: Cosine positive only

Memory tricks:

• "All Students Take Calculus"
• "Add Sugar To Coffee"

Why CAST Works

Think about the signs of x and y coordinates:

• Quadrant I: $(+, +)$ → all positive
• Quadrant II: $(-, +)$ → $\sin = y$ is positive, $\cos = x$ is negative
• Quadrant III: $(-, -)$ → $\tan = \frac{y}{x}$ is positive (negative ÷ negative)
• Quadrant IV: $(+, -)$ → $\cos = x$ is positive, $\sin = y$ is negative

Tips for Mastering the Unit Circle

1. Draw it! Practice sketching the unit circle with all special angles
2. Use symmetry: The circle has symmetry across both axes and both diagonals
3. Start with Quadrant I: Learn those 5 angles perfectly, then use reference angles
4. Remember patterns: The denominators for radians follow a pattern (6, 4, 3, 2)
5. Practice regularly: The unit circle becomes automatic with repetition

Real-World Applications

• Engineering: Analyzing periodic motion and vibrations
• Physics: Projectile motion, waves, oscillations
• Computer graphics: Rotating objects, circular motion
• Music: Sound waves and frequencies
• Astronomy: Planetary orbits and celestial mechanics