Unit Circle Reference

Understanding the Unit Circle

The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) of a coordinate plane. It's a fundamental tool in trigonometry that helps us understand the relationships between angles and their trigonometric functions.

Key Concepts

Coordinates (x, y)

•$x = \cos \theta$
•$y = \sin \theta$
•$x^2 + y^2 = 1$
•$\tan \theta = \frac{y}{x} = \frac{\sin \theta}{\cos \theta}$

Radians vs Degrees

$360° = 2\pi$ rad
$180° = \pi$ rad
$90° = \frac{\pi}{2}$ rad

Quick Calculator

°=rad
rad=°
Common Values
π ≈ 3.14159
30° = π/6 rad
45° = π/4 rad
60° = π/3 rad
90° = π/2 rad
180° = π rad
270° = 3π/2 rad
360° = 2π rad

Quick Convert: rad = deg × (π/180)

Quadrant Rules

ASTC Rule (Sign Rules)

Quadrant I (0-90°)
All positive (+)
Quadrant II (90-180°)
Only Sine positive (+)
Quadrant III (180-270°)
Only Tangent positive (+)
Quadrant IV (270-360°)
Only Cosine positive (+)

Common Values Quick Reference

30° Values:
  • $\sin = \frac{1}{2}$
  • $\cos = \frac{\sqrt{3}}{2}$
  • $\tan = \frac{1}{\sqrt{3}}$
45° Values:
  • $\sin = \frac{\sqrt{2}}{2}$
  • $\cos = \frac{\sqrt{2}}{2}$
  • $\tan = 1$
Point:$(1, 0)$
sin:$0$
cos:$1$
tan:$0$
Point:$(√3/2, 1/2)$
sin:$1/2$
cos:$√3/2$
tan:$√3/3$
Point:$(√2/2, √2/2)$
sin:$√2/2$
cos:$√2/2$
tan:$1$
Point:$(1/2, √3/2)$
sin:$√3/2$
cos:$1/2$
tan:$√3$
Point:$(0, 1)$
sin:$1$
cos:$0$
tan:$\text{undefined}$
Point:$(-1/2, √3/2)$
sin:$√3/2$
cos:$-1/2$
tan:$-√3$
Point:$(-√2/2, √2/2)$
sin:$√2/2$
cos:$-√2/2$
tan:$-1$
Point:$(-√3/2, 1/2)$
sin:$1/2$
cos:$-√3/2$
tan:$-√3/3$
Point:$(-1, 0)$
sin:$0$
cos:$-1$
tan:$0$
Point:$(-√3/2, -1/2)$
sin:$-1/2$
cos:$-√3/2$
tan:$√3/3$
Point:$(-√2/2, -√2/2)$
sin:$-√2/2$
cos:$-√2/2$
tan:$1$
Point:$(-1/2, -√3/2)$
sin:$-√3/2$
cos:$-1/2$
tan:$√3$
Point:$(0, -1)$
sin:$-1$
cos:$0$
tan:$\text{undefined}$
Point:$(1/2, -√3/2)$
sin:$-√3/2$
cos:$1/2$
tan:$-√3$
Point:$(√2/2, -√2/2)$
sin:$-√2/2$
cos:$√2/2$
tan:$-1$
Point:$(√3/2, -1/2)$
sin:$-1/2$
cos:$√3/2$
tan:$-√3/3$

Common Angles Reference Table

Tips and Tricks

Memorization Patterns

  • •Values repeat every 90° with alternating signs
  • •Sine and cosine values are always between -1 and 1
  • •Sine values in Q1 = Cosine values in Q2
  • •Reference angles are always acute (≤ 90°)
ASTC Rule (Sign Rules)
  • Q1 (0-90°): All positive
  • Q2 (90-180°): Only Sine positive
  • Q3 (180-270°): Only Tangent positive
  • Q4 (270-360°): Only Cosine positive

Special Angles

30-60-90 Triangle
$\text{If } \theta = 30°:$
  • $\sin = \frac{1}{2}$
  • $\cos = \frac{\sqrt{3}}{2}$
  • $\tan = \frac{1}{\sqrt{3}}$
45-45-90 Triangle
$\text{If } \theta = 45°:$
  • $\sin = \cos = \frac{\sqrt{2}}{2}$
  • $\tan = 1$

Quick Methods

Finding Reference Angles
  • Q1: θ = θ
  • Q2: θ = 180° - θ
  • Q3: θ = θ - 180°
  • Q4: θ = 360° - θ
Co-function Relationships
  • $\sin(\theta) = \cos(90° - \theta)$
  • $\tan(\theta) = \cot(90° - \theta)$
  • $\sec(\theta) = \csc(90° - \theta)$

Periodicity & Properties

Periods
  • Sine & Cosine:$\text{ Period = }2\pi\text{ or }360°$
  • Tangent:$\text{ Period = }\pi\text{ or }180°$
Even/Odd Functions
  • $\cos(-x) = \cos(x)\text{ (even)}$
  • $\sin(-x) = -\sin(x)\text{ (odd)}$
  • $\tan(-x) = -\tan(x)\text{ (odd)}$