Section 5.1: Angles and Their Measurements

Degree Measure of Angles

Angles are fundamental objects in geometry and trigonometry, representing the rotation between two rays sharing a common endpoint. The degree is a standard unit for measuring angles, based on dividing a circle into 360 equal parts.

• Ray: A part of a line that starts at a point and extends infinitely in one direction.

• Angle: The union of two rays with a common endpoint, called the vertex.

• Initial Side: The fixed ray from which the angle is measured.

• Terminal Side: The ray that is rotated from the initial side to form the angle.

• Central Angle: An angle whose vertex is at the center of a circle; its measure corresponds to the arc it intercepts.

• Standard Position: An angle with its vertex at the origin and its initial side along the positive x-axis in the coordinate plane.

• Degree (°): The measure of an angle corresponding to 1/360 of a full rotation.

• Acute Angle:

• Right Angle:

• Obtuse Angle:

• Straight Angle:

• Quadrantal Angle: An angle whose terminal side lies on one of the coordinate axes.

• Positive/Negative Angles: Positive if measured counterclockwise, negative if measured clockwise.

Note: While degrees can be subdivided into minutes and seconds, this course uses only decimal degrees.

Example:

• An angle of is an acute angle in standard position.

• An angle of is measured clockwise from the positive x-axis.

Coterminal Angles

Coterminal angles share the same initial and terminal sides in standard position, differing by full rotations (multiples of ).

• Definition: Angles and are coterminal if , where is any integer.

Example:

• and are coterminal because .

• and are coterminal because .

Radian Measure of Angles

The radian is another unit for measuring angles, based on the arc length of a circle. It is the standard unit in higher mathematics and science.

• Unit Circle: A circle with radius 1, centered at the origin.

• Radian: The measure of a central angle whose intercepted arc length equals the radius of the circle. One radian is the angle for which the arc length on the unit circle.

• Relationship to Degrees: A full circle is radians, which equals .

Conversion Formula:

• radians

• To convert degrees to radians:

• To convert radians to degrees:

Example:

• Convert to radians: radians.

• Convert radians to degrees: .

Arc Length

The radian measure of a central angle allows for easy calculation of the length of the intercepted arc on a circle.

• Arc Length Formula: For a circle of radius and a central angle (in radians), the arc length is given by:

• This formula arises because the circumference of a circle is , and the fraction of the circle subtended by angle is .

• Unitless Nature: Since radians are defined as a ratio of lengths, they are technically unitless.

Example:

• Find the arc length intercepted by a central angle of $2:

Summary Table: Degree and Radian Measures

Additional info: The above table includes common angles and their radian equivalents, which are frequently used in trigonometry and calculus.