The Unit Circle and periodic functions
The unit circle is a fundamental concept in trigonometry that serves as a bridge between algebraic and geometric representations of trigonometric functions. It is a circle centered at the origin with a radius of 1 unit, which gives it its name.
Understanding the Unit Circle
The unit circle is defined as the set of all points $(x, y)$ that satisfy the equation $x^2 + y^2 = 1$. This equation comes directly from the Pythagorean theorem, as the distance from any point on the circle to the origin is always 1.
When we place this circle on a coordinate plane, we can establish important relationships:
- For any angle $\theta$ (measured counterclockwise from the positive x-axis), we can locate a point $(x, y)$ on the unit circle.
- The x-coordinate of this point equals $\cos(\theta)$
- The y-coordinate of this point equals $\sin(\theta)$
This gives us a geometric interpretation of sine and cosine functions: they are simply the coordinates of points on the unit circle!
Key Points on the Unit Circle
Several points on the unit circle correspond to commonly used angles in trigonometry:
| Angle (radians) | Angle (degrees) | Point on Unit Circle $(\cos\theta, \sin\theta)$ |
|---|---|---|
| 0 | 0° | (1, 0) |
| $\frac{\pi}{6}$ | 30° | $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ |
| $\frac{\pi}{4}$ | 45° | $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ |
| $\frac{\pi}{3}$ | 60° | $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ |
| $\frac{\pi}{2}$ | 90° | (0, 1) |