Concepts · 3 min read
The Unit Circle and Trigonometry Basics (Degrees and Radians)
Sine and cosine as coordinates
Draw a circle of radius 1 centred at the origin. Rotate a point counter-clockwise by angle θ from the positive x-axis. Its x-coordinate is cos θ and its y-coordinate is sin θ. Because the radius is 1, x² + y² = 1, which is sin²θ + cos²θ = 1.
Key angles
| Degrees | Radians | cos θ | sin θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5236 | 0.866025403784 | 0.5 |
| 45° | 0.7854 | 0.707106781187 | 0.707106781187 |
| 60° | 1.0472 | 0.5 | 0.866025403784 |
| 90° | 1.5708 | 0 | 1 |
| 120° | 2.0944 | -0.5 | 0.866025403784 |
| 135° | 2.3562 | -0.707106781187 | 0.707106781187 |
| 150° | 2.618 | -0.866025403784 | 0.5 |
| 180° | 3.1416 | -1 | 0 |
Radians
A radian is the angle whose arc length equals the radius. 180° = π radians, so multiply degrees by π/180. A full turn is 2π ≈ 6.2832.
Degree mode: sin 45° + cos 45° = √2
Order-of-operations steps 3 steps
Start: sin(45) + cos(45)
- Function
sin(45) = 0.7071067812 (angle in degrees)→ 0.7071067812 + cos(45) - Function
cos(45) = 0.7071067812 (angle in degrees)→ 0.7071067812 + 0.7071067812 - Add
0.7071067812 + 0.7071067812 = 1.414213562→ 1.414213562
Use it
Switch DEG/RAD in the scientific calculator, plot waves in the graphing calculator, and keep the trigonometry cheat sheet handy.
Frequently asked questions
Why do mathematicians prefer radians?
Formulas such as the derivative of sin x = cos x are simplest when angles are in radians.
What is cos 90°?
0, because the point is at (0, 1).
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