Measuring Angles

Degrees and Radians

Angles can be measured in two primary units: degrees and radians. Understanding how to convert between these units is fundamental in trigonometry.

• Degree: A full circle is 360 degrees.

• Radian: A full circle is radians. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

Conversion Formulas:

• Degrees to radians:

• Radians to degrees:

Example: Convert to radians:

• radians

Arc Length and Sector Area

Arc Length

The length of an arc of a circle is determined by the radius and the angle (in radians) subtended by the arc at the center.

• Formula:

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

Example: Find the arc length for a circle of radius 5 and angle radians:

Sector Area

The area of a sector of a circle is a fraction of the area of the whole circle, determined by the angle in radians.

• Formula:

• Where is the area, is the radius, and is the angle in radians.

Example: Find the area of a sector with radius 4 and angle radians:

Applications: Linear and Angular Speed

Definitions

• Linear Speed (): The rate at which distance is covered along a circular path.

• Angular Speed (): The rate at which the angle changes, measured in radians per unit time.

Example: A water sprinkler rotates at $30 feet. Find the linear speed at the tip of the arm.

• First, find angular speed: radians/minute

• Then, feet/minute

Summary Table: Angle and Arc Relationships