Radian Measure and the Unit Circle
Define radian measure and convert between degrees and radians. Plot all key angles (multiples of π/6 and π/4) on the unit circle and find exact (x, y) coordinates using special right triangle ratios.
A lesson of Algebra 2, in Introduction to Trigonometry, about 10 minutes.
By the end of this lesson, you will be able to convert between degree and radian measure, locate standard angle positions on the unit circle, and find exact coordinates for key angles without a calculator.
The Big Idea
Radian measure is the natural unit for angles. Unlike degrees (which are arbitrary), radians tie angle measure directly to arc length: one radian is the angle that intercepts an arc equal in length to the radius. This makes radians essential for calculus, physics, and higher mathematics — all trigonometric derivatives and integrals are stated in radians.
Definition of Radian
One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
$$\theta \text{ (radians)} = \frac{\text{arc length}}{\text{radius}} = \frac{s}{r}$$
Since the circumference of a circle is $2\pi r$, a full rotation ($360°$) equals $2\pi$ radians.
$$360° = 2\pi \text{ rad} \implies 180° = \pi \text{ rad}$$
Conversion Formulas
Degrees to radians: multiply by $\dfrac{\pi}{180°}$
Radians to degrees: multiply by $\dfrac{180°}{\pi}$
Key Angle Conversions
| Degrees | Radians |
|---|---|
| 0° | 0 |
| 30° | $\pi/6$ |
| 45° | $\pi/4$ |
| 60° | $\pi/3$ |
| 90° | $\pi/2$ |
| 120° | $2\pi/3$ |
| 135° | $3\pi/4$ |
| 150° | $5\pi/6$ |
| 180° | $\pi$ |
| 270° | $3\pi/2$ |
| 360° | $2\pi$ |
The Unit Circle
The unit circle is the circle of radius 1 centered at the origin: $x^2 + y^2 = 1$.
For angle $\theta$ measured from the positive $x$-axis, the point on the unit circle is $(\cos\theta, \sin\theta)$.
This definition extends sine and cosine beyond acute angles to all real numbers.