Angles and Their Measurement

Standard Position and Rotation

An angle is in standard position if its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis. The direction of rotation determines the sign of the angle:

• Positive angles are generated by counterclockwise rotation.

• Negative angles are generated by clockwise rotation.

Quadrants and Quadrantal Angles

The terminal side of an angle in standard position determines its quadrant:

• If the terminal side lies in Quadrant II, the angle is between 90° and 180°.

• If the terminal side lies in Quadrant IV, the angle is between 270° and 360° (or negative between 0° and -90°).

• If the terminal side lies on an axis, the angle is called a quadrantal angle.

Examples: Drawing Angles in Standard Position

• 135°: Lies in Quadrant II.

• –180°: Negative rotation, terminal side on the negative x-axis.

• 90°: Terminal side on the positive y-axis (quadrantal angle).

• 495°: Equivalent to 135° after subtracting 360° (one full revolution), but represents one extra revolution.

Additional info: Angles differing by multiples of 360° (or 2π radians) are called coterminal angles.

Radians and Degree Measure

Definition of a Radian

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

Relationship Between Degrees and Radians

• One full revolution around a circle is radians or 360°.

• To convert degrees to radians, multiply by .

• To convert radians to degrees, multiply by .

Example: Convert 60° to radians:

• radians

Example: Convert 2 radians to degrees:

Common Angles Table

The following table lists common angles in both degrees and radians:

Arc Length and Sector Area

Arc Length

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

Example: Find the arc length of a circle of radius 3 meters subtended by a central angle of $2$ radians.

• meters

Sector Area

The area of a sector of a circle of radius formed by a central angle (in radians) is:

Example: A water sprinkler sprays water over a distance of 36 feet while rotating through an angle of 140°. What area of lawn receives water?

• Convert 140° to radians: radians

• Area: ft2

Circular Motion: Linear and Angular Speed

Linear Speed

The linear speed of an object moving along a circle of radius is the distance traveled per unit time:

Example: A child spins a rock at the end of a 2-foot rope at 180 revolutions per minute. Find the linear speed when released.

• Distance per revolution: feet

• Total distance per minute: feet

• Linear speed: feet/minute

Angular Speed

The angular speed (in radians per unit time) is the angle swept out per unit time:

Relationship Between Linear and Angular Speed

Linear speed and angular speed are related by the formula:

Additional info: This formula only holds when is measured in radians per unit time. If using degrees, a conversion factor is required.