Angles and Their Measure

Introduction to Angles

Angles are fundamental geometric objects formed by two rays sharing a common endpoint called the vertex. In trigonometry and precalculus, angles are used to describe rotation and position in a plane.

• 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.

• Positive Angle: Formed by counterclockwise rotation from the initial side.

• Negative Angle: Formed by clockwise rotation from the initial side.

Example: Drawing a 45° angle involves rotating the terminal side 45° counterclockwise from the initial side.

Angles in Standard Position

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.

• Quadrantal Angles: Angles whose terminal side lies on the x-axis or y-axis (e.g., 0°, 90°, 180°, 270°).

• Angles in Quadrants: The terminal side of an angle may lie in any of the four quadrants, depending on its measure.

Example: A 135° angle has its terminal side in the second quadrant.

Measuring Angles: Degrees and Radians

Degrees

The degree is a unit of angular measure. One full revolution is 360 degrees.

• 1 revolution = 360°

• Quadrantal Angles: 0°, 90°, 180°, 270°, 360°

Example: A 90° angle is a right angle, with its terminal side along the positive y-axis.

Radians

The radian is another unit of angular measure, based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.

• 1 revolution = 2π radians

• Relationship:

Example: An angle of π/2 radians is equivalent to 90°.

Converting Between Degrees and Radians

To convert between degrees and radians, use the following formulas:

• Degrees to Radians:

• Radians to Degrees:

Example: Convert 150° to radians:

radians

Example: Convert radians to degrees:

Common Degree-Radian Conversions Table

Arc Length and Area of a Sector

Arc Length

The length of an arc of a circle is proportional to the radius and the angle (in radians) subtended by the arc.

• Formula:

• s: Arc length

• r: Radius of the circle

• θ: Central angle in radians

Example: Find the arc length of a circle with radius 4.05 m and central angle 0.87 radians:

meters

Area of a Sector

The area of a sector of a circle is a fraction of the area of the entire circle, determined by the central angle (in radians).

• Formula:

• A: Area of the sector

• r: Radius of the circle

• θ: Central angle in radians

Example: Find the area of a sector with radius 6 inches and central angle 30° (convert to radians first):

square inches

Practice Problems

• Draw angles of various measures in standard position.

• Convert between degrees and radians.

• Find arc length and area of a sector for given radius and angle.

Summary Table: Key Formulas

Additional info: These notes cover foundational trigonometric concepts from Precalculus Chapter 5 (Trigonometric Functions), specifically the measurement of angles, conversion between units, and geometric applications involving circles.