What Is the Unit Circle?
The Unit Circle is a circle with a radius of exactly 1 unit (r = 1) centered at the origin (0, 0) of the Cartesian coordinate plane. It is the cornerstone of modern trigonometry because it extends angle measurement beyond 90-degree right triangles to all positive and negative angles.
The algebraic equation of the unit circle is:
x² + y² = 1
The Fundamental Trigonometric Identity on the Unit Circle
For any angle θ measured counterclockwise from the positive x-axis, the terminal ray intersects the unit circle at point (x, y), where:
- x = cos(θ) (The horizontal distance is the Cosine)
- y = sin(θ) (The vertical distance is the Sine)
- tan(θ) = y / x = sin(θ) / cos(θ) (The slope of the ray is the Tangent, where x ≠ 0)
Special Reference Angles (Quadrant I)
| Angle (Degrees) | Angle (Radians) | cos(θ) [x] | sin(θ) [y] | tan(θ) [y/x] |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3 / 2 | 1/2 | 1 / √3 = √3/3 |
| 45° | π/4 | √2 / 2 | √2 / 2 | 1 |
| 60° | π/3 | 1/2 | √3 / 2 | √3 |
| 90° | π/2 | 0 | 1 | Undefined |
The ASTC Rule for Quadrant Signs
To determine the signs (+ or -) of trig functions in all 4 quadrants, remember the mnemonic "All Students Take Calculus":
- Quadrant I (0° to 90°): All functions (sin, cos, tan) are positive (+, +).
- Quadrant II (90° to 180°): Sine only is positive (-, +).
- Quadrant III (180° to 270°): Tangent only is positive (-, -).
- Quadrant IV (270° to 360°): Cosine only is positive (+, -).
Step-by-Step Example: Find Exact Value of sin(210°) and cos(210°)
- Locate the Quadrant: 210° lies between 180° and 270°, so it is in Quadrant III.
- Find the Reference Angle: In Quadrant III, Reference Angle = 210° - 180° = 30°.
- Look up 30° values: cos(30°) = √3/2, sin(30°) = 1/2.
- Apply Quadrant III Signs: In Quadrant III, both x and y are negative.
- Conclusion: cos(210°) = -√3/2, and sin(210°) = -1/2.