Angles
A ray is part of a line with one endpoint that extends infinitely in one direction.
The ray AB
An angle is the figure formed by two distinct rays with a common endpoint.
The angle ∠BAC
By designating one ray as the initial side and the other as the terminal side, an angle can be interpreted as the rotation that carries the initial side to the terminal side. The common endpoint of the rays is called the vertex of the angle.
The measure of an angle is the amount of rotation required to move the initial side to the terminal side.
We follow the convention that if the rotation is counterclockwise, then the angle measure is positive; if the rotation is clockwise, then the angle measure is negative.
Degree measure
A rotation of a ray through one full turn back onto itself is called a complete rotation. By convention, a complete rotation is divided into 360 equal parts, and a degree is defined as
1° = 1360 of a complete rotation.
| Angle | Measure | Example |
|---|
| Acute | 0° < θ < 90° | 45° |
| Right | θ = 90° | 90° |
| Obtuse | 90° < θ < 180° | 120° |
| Straight | θ = 180° | 180° |
| Reflex | 180° < θ ≤ 360° | 330° |
A negative angle may have measure −135°.
The same symbol is used to denote an angle and its measure, typically a lowercase Greek letter such as θ.
θ = 360°θ = 1°
Minutes and seconds
A minute is defined as
1′ = 160 of a degree.
A second is defined as
1″ = 160 of a minute.
Example: 15° 30′ 15″ ≈ 15.504167°.
Radian measure
Given a fixed circle, a central angle is an angle whose vertex is at the center of the circle and whose sides are radii of the circle.
One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.
θ = 2 radθ = 2π rad
Let s be the length of the arc intercepted by a central angle θ in radians. From the definition of the radian, we obtain the formula
s = rθ.
Conversion of angle measures
If no unit (degrees or radians) is indicated, an angle is assumed to be measured in radians.
Degrees to radians
Multiply by π180
Radians to degrees
Multiply by 180π
Convert the angle measures
| Degrees | Radians |
|---|
| 45° | |
| 120° | |
| π5 |
| 3π4 |
| 270° | |
Angles in standard position
An angle is in standard position if its vertex is at the origin (0, 0) and its initial side lies along the positive x-axis.
An angle in standard position whose terminal side lies in a quadrant is said to lie in the quadrant containing its terminal side.
Coterminal angles
Two angles in standard position with coincident terminal sides are called coterminal angles.
- Find two different angles in standard position coterminal to 2π5.
- Find an angle coterminal to 19π6 such that 0 ≤ θ < 2π.
- Find an angle coterminal to −13π3 such that 0 ≤ θ < 2π.
Quadrantal angles
An angle in standard position whose terminal side lies on a coordinate axis is called a quadrantal angle.