Angles and Their Measurements: Degree and Radian Measure

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: $0^\circ < \alpha < 90^\circ$

• Right Angle: $90^\circ$

• Obtuse Angle: $90^\circ < \alpha < 180^\circ$

• Straight Angle: $180^\circ$

• 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 $45^\circ$ is an acute angle in standard position.

• An angle of $-120^\circ$ 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 $360^\circ$).

• Definition: Angles $\alpha$ and $\beta$ are coterminal if $m(\beta) = m(\alpha) + k \cdot 360^\circ$, where $k$ is any integer.

Example:

• $30^\circ$ and $390^\circ$ are coterminal because $390^\circ = 30^\circ + 360^\circ$.

• $-45^\circ$ and $315^\circ$ are coterminal because $315^\circ = -45^\circ + 360^\circ$.

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 $s = 1$ on the unit circle.

• Relationship to Degrees: A full circle is $2\pi$ radians, which equals $360^\circ$.

Conversion Formula:

• $180^\circ = \pi$ radians

• To convert degrees to radians: $\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$

• To convert radians to degrees: $\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}$

Example:

• Convert $90^\circ$ to radians: $90^\circ \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.

• Convert $\frac{2\pi}{3}$ radians to degrees: $\frac{2\pi}{3} \times \frac{180}{\pi} = 120^\circ$.

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 $r$ and a central angle $\alpha$ (in radians), the arc length $s$ is given by:

$s = r \alpha$

• This formula arises because the circumference of a circle is $2\pi r$, and the fraction of the circle subtended by angle $\alpha$ is $\frac{\alpha}{2\pi}$.

• 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 $2$radians in a circle of radius $5$:

$s = r \alpha = 5 \times 2 = 10$

Summary Table: Degree and Radian Measures

Angle (Degrees)	Angle (Radians)	Type
0°	0	Zero Angle
30°	$\frac{\pi}{6}$	Acute
45°	$\frac{\pi}{4}$	Acute
60°	$\frac{\pi}{3}$	Acute
90°	$\frac{\pi}{2}$	Right
180°	$\pi$	Straight
270°	$\frac{3\pi}{2}$	Obtuse/Quadrantal
360°	$2\pi$	Full Rotation

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