Angles and Their Measurement

Definition and Components of an Angle

An angle is formed by two rays sharing a common endpoint, called the vertex. The ray from which the angle begins is the initial side, and the ray where the angle ends is the terminal side.

• Vertex: The common endpoint of the two rays.

• Initial Side: The starting position of the ray.

• Terminal Side: The position of the ray after rotation.

Angles in Standard Position

An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis. The terminal side is then rotated from the initial side to form the angle.

Positive and Negative Angles

Angles are classified by the direction of rotation from the initial side to the terminal side:

• Positive angles: Generated by counterclockwise rotation.

• Negative angles: Generated by clockwise rotation.

Quadrantal Angles

A quadrantal angle is an angle whose terminal side lies on either the x-axis or the y-axis. These angles are important reference points in trigonometry.

Measuring Angles: Degrees and Radians

Degree Measure

The degree is a common unit for measuring angles. A full rotation around a circle is 360 degrees (360°).

• Acute angle: Less than 90°

• Right angle: Exactly 90°

• Obtuse angle: Greater than 90° but less than 180°

• Straight angle: Exactly 180°

Radian Measure

The radian is the standard unit of angular measure in mathematics. One radian is the measure of a central angle that intercepts an arc equal in length to the radius of the circle.

• Central angle: An angle whose vertex is at the center of a circle.

• Radian definition: $1$ radian is the angle subtended by an arc equal to the radius of the circle.

Arc Length and Radian Measure

The measure of a central angle in radians is given by the ratio of the arc length to the radius:

• Formula:

• Where is the angle in radians, is the arc length, and is the radius.

Example: Computing Radian Measure

• Given: A central angle in a circle of radius 12 feet intercepts an arc of length 42 feet.

• Solution: radians.

Conversion Between Degrees and Radians

The relationship between degrees and radians is fundamental in trigonometry:

• radians

• To convert degrees to radians:

• To convert radians to degrees:

Examples: Degree-Radian Conversion

• Convert to radians: radians

• Convert radians to degrees:

Drawing Angles in Standard Position

Visualizing Angles

Angles in standard position can be visualized by rotating the terminal side from the positive x-axis by a specified amount, either clockwise (negative) or counterclockwise (positive).

• Example: is a negative angle, rotated clockwise by of a revolution.

• Example: is a positive angle, rotated counterclockwise by of a revolution.

Common Angles in Trigonometry

Many angles frequently used in trigonometry correspond to simple fractions of a full revolution. The table below summarizes these relationships:

Coterminal Angles

Definition and Properties

Coterminal angles are angles in standard position that share the same initial and terminal sides but may have different measures due to multiple rotations.

• In degrees: is coterminal with , where is any integer.

• In radians: is coterminal with , where is any integer.

Examples: Finding Coterminal Angles

• Find a positive coterminal angle for :

• Find a positive coterminal angle for :

• Find a positive coterminal angle for :

Arc Length and Circular Motion

Arc Length Formula

The length of an arc, , intercepted by a central angle (in radians) in a circle of radius is given by:

• Formula:

Example: Finding Arc Length

• Given: inches, radians

• Arc length: inches

Linear and Angular Speed

Definitions

• Linear speed (): The rate at which a point moves along a circular path. , where is arc length and is time.

• Angular speed (): The rate at which the central angle changes. , where is in radians.

Relationship Between Linear and Angular Speed

• Formula:

• Where is the radius and is the angular speed in radians per unit time.

Example: Finding Linear Speed

• A 45-rpm record has an angular speed of radians per minute.

• If the needle is 1.5 inches from the center, inches per minute.