Angles, Degree and Radian Measure, and Coterminal Angles

Angles and Their Measurement

Definition of an Angle

An angle is formed by two rays that share a common endpoint called the vertex. The ray that remains fixed is called the initial side, and the ray that is rotated is called the terminal side of the angle.

• Positive angles are generated by rotating the terminal side counterclockwise from the initial side.

• Negative angles are generated by rotating the terminal side clockwise from the initial side.

Standard Position of an Angle

An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side may lie in any quadrant or on an axis.

Quadrants

The coordinate plane is divided into four quadrants, labeled I, II, III, and IV, in a counterclockwise direction starting from the positive x-axis.

Degree Measure

Understanding Degree Measure

Angles can be measured in degrees, where one complete revolution is 360°. Common degree measures include:

• 360°: One complete counterclockwise revolution.

• 90°: One-fourth of a counterclockwise revolution.

• -45°: One-eighth of a clockwise revolution (negative direction).

Examples of Angles in Degrees

Angles such as 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 225°, 270°, 315°, and 360° are commonly used and can be visualized on the coordinate plane. Negative angles, such as -30°, -45°, -60°, -90°, -120°, -135°, -150°, -180°, -225°, -270°, -315°, and -360°, are measured clockwise.

Coterminal Angles

Definition and Calculation

Coterminal angles are angles that share the same initial and terminal sides but may have different measures. Coterminal angles can be found by adding or subtracting integer multiples of 360° (or $2 ext{π}$ radians) to a given angle.

• For any angle $heta$, coterminal angles are given by $heta + 360^ ext{o} imes n$, where $n$ is any integer.

• Every angle has a coterminal angle of least non-negative measure, denoted as $Q_ heta$.

Important Note: Coterminal angles are not equal, even though they share the same terminal side.

Examples of Coterminal Angles in Degrees

• $45^ ext{o}$ and $405^ ext{o}$ are coterminal.

• $-315^ ext{o}$ is coterminal with $45^ ext{o}$.

Radian Measure

Definition of a Radian

A radian is the measure of a central angle whose intercepted arc is equal in length to the radius of the circle.

• On a circle of radius $r$, an angle of 1 radian subtends an arc of length $r$.

Relationship Between Degrees and Radians

The relationship between degrees and radians is given by:

• $360^ ext{o} = 2 ext{π}$ radians

• $180^ ext{o} = ext{π}$ radians

Conversion Formulas

• To convert degrees to radians: $1^ ext{o} = rac{ ext{π}}{180}$ radians

• To convert radians to degrees: $1$radian$= rac{180^ ext{o}}{ ext{π}}$

Examples of Angles in Radians

• $2 ext{π}$ radians: One complete counterclockwise revolution.

• $rac{ ext{π}}{2}$ radians: One-fourth of a counterclockwise revolution.

• $- rac{ ext{π}}{4}$ radians: One-eighth of a clockwise revolution.

Common Angles in Radians

Angles such as $rac{ ext{π}}{6}$, $rac{ ext{π}}{4}$, $rac{ ext{π}}{3}$, $rac{ ext{π}}{2}$, $rac{2 ext{π}}{3}$, $rac{3 ext{π}}{4}$, $ext{π}$, $rac{5 ext{π}}{6}$, $rac{7 ext{π}}{6}$, $rac{5 ext{π}}{4}$, $rac{4 ext{π}}{3}$, $rac{3 ext{π}}{2}$, $rac{5 ext{π}}{3}$, $rac{7 ext{π}}{4}$, and $2 ext{π}$ are commonly used. Negative angles are measured clockwise, such as $- rac{ ext{π}}{6}$, $- rac{ ext{π}}{4}$, $- rac{ ext{π}}{3}$, $- rac{ ext{π}}{2}$, etc.

Coterminal Angles in Radians

Definition and Calculation

Coterminal angles in radians are found by adding or subtracting integer multiples of $2 ext{π}$ to a given angle.

• For any angle $heta$, coterminal angles are given by $heta + 2 ext{π} imes n$, where $n$ is any integer.

• Every angle has a coterminal angle of least non-negative measure, denoted as $Q_ heta$.

Important Note: Coterminal angles are not equal, even though they share the same terminal side.

Examples of Coterminal Angles in Radians

• $rac{ ext{π}}{3}$ and $rac{7 ext{π}}{3}$ are coterminal.

• $- rac{5 ext{π}}{3}$ is coterminal with $rac{ ext{π}}{3}$.

Summary Table: Degree and Radian Conversion

Degrees	Radians
$360^ ext{o}$	$2 ext{π}$
$180^ ext{o}$	$ext{π}$
$90^ ext{o}$	$rac{ ext{π}}{2}$
$45^ ext{o}$	$rac{ ext{π}}{4}$
$30^ ext{o}$	$rac{ ext{π}}{6}$
$60^ ext{o}$	$rac{ ext{π}}{3}$

Key Points and Warnings

• Do not equate coterminal angles: For example, $45^ ext{o} eq -315^ ext{o}$, even though they are coterminal.

• Similarly, $rac{2 ext{π}}{4} eq rac{3 ext{π}}{4}$, even though they may be coterminal.

Additional info:

• Angles are fundamental in trigonometry and are used to describe rotation, direction, and periodic phenomena.

• Understanding coterminal angles is essential for solving trigonometric equations and analyzing periodic functions.