In trigonometry and calculus, angles are almost always measured in radians rather than degrees. Understanding radian measure, and knowing how to convert between the two systems, is the essential first step before working with any trigonometric function.
Quick Reference
| Formula | Name | Notes |
|---|---|---|
| $2\pi \text{ rad} = 360°$ | Radian-degree conversion | Full circle |
| $\theta(\text{rad}) = \dfrac{\pi}{180}\,\theta(°)$ | Degrees to radians | Multiply by π/180 |
| $\theta(°) = \dfrac{180}{\pi}\,\theta(\text{rad})$ | Radians to degrees | Multiply by 180/π |
| s = rθ | Arc Length | θ must be in radians |
| $A = \dfrac{1}{2}r^2\theta$ | Area of a Sector | θ must be in radians |
Radian Measure
The most common way to measure angles is in degrees: a right angle is 90° and a full rotation is 360°. However, there is a more natural unit for calculus called the radian.
To measure an angle in radians, place its vertex at the center of a unit circle (a circle of radius 1). The angle cuts out an arc on the circumference. The length of that arc, in units matching the radius, is the measure of the angle in radians.
Since the full circumference of a unit circle is 2π, a complete rotation equals 2π radians:
From this we get the two conversion factors:
$ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \approx 57.296° $$ 1° = \frac{\pi}{180} \text{ radians} \approx 0.01745 \text{ rad} $Converting degrees to radians: multiply by $\dfrac{\pi}{180}$.
Converting radians to degrees: multiply by $\dfrac{180}{\pi}$.
Example. Convert 45° to radians.
Solution
$ \frac{\pi}{180} \cdot 45 = \frac{45\pi}{180} = \frac{\pi}{4} \text{ rad} $Important convention: In mathematics, angles are measured in radians by default unless the degree symbol ° is shown explicitly. So $\sin 1$ means the sine of 1 radian, and $\sin 1°$ means the sine of 1 degree. These are very different numbers:
sin 1 ≈ 0.841471 but sin 1° ≈ 0.017452When you see an angle like π/6, it means π/6 radians, which equals 30°, not π/6 degrees.
Common Angle Conversions
| Degrees | Radians |
|---|---|
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 120° | 2π/3 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
Arc Length
An arc on a circle subtends a central angle at the center. If you know the angle in radians and the radius, you can find the arc length directly.
Arc Length Theorem. If an arc subtends a central angle of θ radians on a circle of radius r, then the length s of the arc is:
s = rθ (θ in radians)
Derivation (click to expand)
The arc length s is the same fraction of the full circumference 2πr as the angle θ is of a full revolution 2π radians: $ \frac{s}{2\pi r} = \frac{\theta}{2\pi} $ Solving for s gives s = rθ.Example. Find the length of the arc intercepted by a central angle of 50° on a circle of radius 36 in.
Solution
First convert 50° to radians: $ \theta = \frac{\pi}{180} \cdot 50 = \frac{5\pi}{18} \text{ rad} $ Then apply the arc length formula: $ s = r\theta = 36 \cdot \frac{5\pi}{18} = 10\pi \approx 31.4 \text{ in} $
Area of a Circular Sector
A circular sector is the region bounded by two radii and the arc between them (like a "pie slice").
Area of a Sector. The area A of a sector with central angle θ (in radians) in a circle of radius r is:
$ A = \frac{1}{2}r^2\theta \qquad (\theta \text{ in radians}) $The unit of A is the square of the unit used for r (for example, square inches if r is in inches).
Derivation (click to expand)
The area of the full disk is π r². A sector with angle θ is the fraction θ/(2π) of the full disk: $ A = \pi r^2 \cdot \frac{\theta}{2\pi} = \frac{1}{2}r^2\theta $Example. For a circle of radius 18 in, find the area of a sector intercepted by a central angle of 70°.
Solution
Convert 70° to radians: $ \theta = \frac{\pi}{180} \cdot 70 = \frac{7\pi}{18} \text{ rad} $ Then apply the sector area formula: $ A = \frac{1}{2}r^2\theta = \frac{1}{2}(18)^2 \cdot \frac{7\pi}{18} = \frac{1}{2} \cdot 18 \cdot 7\pi = 63\pi \approx 197.9 \text{ in}^2 $
Directed Angles
Any angle has two sides. To measure rotations, we designate one side as the initial side and the other as the terminal side. We think of an angle as being generated by the rotation of the initial side to the terminal side. If the rotation is counterclockwise, the angle is positive; if the rotation is clockwise, the angle is negative. We allow angles of more than one complete rotation.
 |
 |
| (a) A positive angle in standard position | (b) A negative angle in standard position |
Angles in Standard Position
An angle is in its standard position if its vertex is at the origin and its initial side lies along the positive x-axis. We say an angle is a first-quadrant angle, or is in the first quadrant, if in standard position its terminal side lies in Quadrant I. Similar definitions apply for the second, third, and fourth quadrants.
For example, $\pi/4$, −315°, and 405° are first-quadrant angles — their terminal sides all land in Quadrant I. Note that these three angles are coterminal: −315° + 360° = 45° and 405° − 360° = 45°, so after accounting for full rotations, all three reach the same terminal side.
Similarly, $5\pi/6$ is a second-quadrant angle. Since $\pi/2 < 5\pi/6 < \pi$, the terminal side lies between the positive y-axis and the negative x-axis.
Finally, −60° and 300° are fourth-quadrant angles. They are coterminal: −60° + 360° = 300°, and since 270° < 300° < 360°, the terminal side lies in Quadrant IV.
