Angles and Radian Measure
Angles and Radian Measure
Trigonometry begins with angles. An angle is formed by two rays sharing a common endpoint (the vertex), and it can be measured in degrees or radians. Degrees divide a full rotation into 360 equal parts, while radians relate the angle directly to the arc length on a unit circle.
Radian measure may feel unfamiliar at first, but it is the standard in calculus and physics because it simplifies formulas — for example, the arc length formula s = rθ works only when θ is in radians.
This lesson covers angle terminology, degree and radian conversions, coterminal angles, and complementary and supplementary angles, giving you the vocabulary and tools for everything that follows.
An angle is formed by two rays sharing a common endpoint called the vertex. The amount of rotation from the initial side to the terminal side determines the angle's measure.
Degree Measure
A full rotation is $360°$. A right angle is $90°$. A straight angle is $180°$.
Radian Measure
One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. A full rotation is $2\pi$ radians.
$$\text{Radians} = \text{Degrees} \times \frac{\pi}{180°}$$
$$\text{Degrees} = \text{Radians} \times \frac{180°}{\pi}$$
Common Angle Conversions
| Degrees | Radians |
|---|---|
| $30°$ | $\dfrac{\pi}{6}$ |
| $45°$ | $\dfrac{\pi}{4}$ |
| $60°$ | $\dfrac{\pi}{3}$ |
| $90°$ | $\dfrac{\pi}{2}$ |
| $120°$ | $\dfrac{2\pi}{3}$ |
| $180°$ | $\pi$ |
| $270°$ | $\dfrac{3\pi}{2}$ |
| $360°$ | $2\pi$ |
Convert $150°$ to radians.
$$150° \times \frac{\pi}{180°} = \frac{150\pi}{180} = \frac{5\pi}{6}$$
Convert $\dfrac{7\pi}{4}$ to degrees.
$$\frac{7\pi}{4} \times \frac{180°}{\pi} = \frac{7 \times 180°}{4} = 315°$$
Convert $-210°$ to radians.
$$-210° \times \frac{\pi}{180°} = -\frac{7\pi}{6}$$
Arc Length and Area of a Sector
For a circle of radius $r$ and central angle $\theta$ (in radians):
$$s = r\theta \qquad A = \tfrac{1}{2}r^2\theta$$
Find the arc length subtended by a central angle of $\dfrac{\pi}{3}$ in a circle of radius 12 cm.
$$s = 12 \cdot \frac{\pi}{3} = 4\pi \approx 12.57 \text{ cm}$$
Practice Problems
Show Answer Key
1. $\dfrac{5\pi}{4}$
2. $300°$
3. $-\dfrac{\pi}{4}$
4. $330°$
5. $\dfrac{2\pi}{5}$
6. $s = 10 \cdot \frac{2\pi}{5} = 4\pi \approx 12.57$
7. $A = \frac{1}{2}(64)\frac{\pi}{4} = 8\pi \approx 25.13$
8. $3\pi$
9. $54°$
10. $s = 14 \cdot \frac{5\pi}{7} = 10\pi \approx 31.42$ in.
11. $\frac{\pi}{180} \approx 0.01745$ rad
12. $-240°$