18.1 Angles of Rotation and Radian Measure

An angle of rotation is defined in trigonometry as an angle formed by the starting and ending positions of a ray that rotates about its endpoint.
An angle is considered to be in standard position within a coordinate plane when:
- The starting position of the ray, known as the initial side, lies on the positive $x$ -axis.
- The endpoint of the ray is located at the origin $(0, 0)$ .
A curved arrow is utilized to indicate the amount and direction of rotation, leading to the terminal side (the ending position) of the angle.
Rotation Directions and Measures:
- Counterclockwise: Rotations in this direction result in positive angle measures.
- Clockwise: Rotations in this direction result in negative angle measures.

Coterminal Angles: These are angles that share the same terminal side despite having different measures or directions of rotation.
- Example: An angle of $257^\circ$ (counterclockwise) and an angle of $-103^\circ$ (clockwise) are coterminal because they land on the same ray position.
Single Revolution Bounds: If a rotation is less than one full revolution in a counterclockwise direction, its measure $\theta$ falls between $0^\circ$ and $360^\circ$ .
Multiple Revolutions: Because a ray can undergo any number of revolutions, the coordinate plane can represent:
- Any positive real number (corresponding to counterclockwise rotation).
- Any negative real number (corresponding to clockwise rotation).
- Example: A rotation of more than one but less than two full counterclockwise revolutions results in a measure between $360^\circ$ and $720^\circ$ . An example provided is $420^\circ$ .

Standard formulas are used to convert between the two primary units of angular measure:
- Converting Degrees to Radians: Multiply the number of degrees by $\frac{\pi \text{ radians}}{180^\circ}$ .
- Converting Radians to Degrees: Multiply the number of radians by $\frac{180^\circ}{\pi \text{ radians}}$ .

To convert $20^\circ$ : $20^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{9}$ .
To convert $315^\circ$ : $315^\circ \times \frac{\pi}{180^\circ} = \frac{7\pi}{4}$ .
To convert $600^\circ$ : $600^\circ \times \frac{\pi}{180^\circ} = \frac{10\pi}{3}$ .
To convert $-60^\circ$ : $-60^\circ \times \frac{\pi}{180^\circ} = -\frac{\pi}{3}$ .
To convert $-540^\circ$ : $-540^\circ \times \frac{\pi}{180^\circ} = -3\pi$ .
To convert $70^\circ$ : $70^\circ \times \frac{\pi}{180^\circ} = \frac{7\pi}{18}$ .
To convert $-270^\circ$ : $-270^\circ \times \frac{\pi}{180^\circ} = -\frac{3\pi}{2}$ .
To convert $-945^\circ$ : $-945^\circ \times \frac{\pi}{180^\circ} = -\frac{21\pi}{4}$ .
To convert $2160^\circ$ : $2160^\circ \times \frac{\pi}{180^\circ} = 12\pi$ .

To convert $\frac{\pi}{8}$ : $\frac{\pi}{8} \times \frac{180^\circ}{\pi} = 22.5^\circ$ .
To convert $\frac{4\pi}{3}$ : $\frac{4\pi}{3} \times \frac{180^\circ}{\pi} = 240^\circ$ .
To convert $\frac{9π}{2}$ : $\frac{9π}{2} \times \frac{180^\circ}{\pi} = 810^∘$ .
To convert $-\frac{7\pi}{12}$ : $-\frac{7\pi}{12} \times \frac{180^\circ}{\pi} = -105^\circ$ .
To convert $-\frac{13\pi}{6}$ : $-\frac{13\pi}{6} \times \frac{180^\circ}{\pi} = -390^\circ$ .
To convert $\frac{33\pi}{18}$ : $\frac{33\pi}{18} \times \frac{180^\circ}{\pi} = 330^\circ$ .
To convert $\frac{11\pi}{4}$ : $\frac{11\pi}{4} \times \frac{180^\circ}{\pi} = 495^\circ$ .
To convert $-\frac{5\pi}{3}$ : $-\frac{5\pi}{3} \times \frac{180^∘}{\pi} = -300^∘$ .
To convert $-\frac{7\pi}{2}$ : $-\frac{7\pi}{2} \times \frac{180^∘}{\pi} = -630^∘$ .

Example: $84^\circ$
- Nearest positive coterminals: $84^\circ + 360^\circ = 444^\circ$ and $84^\circ + 2(360^\circ) = 804^\circ$ .
- Nearest negative coterminals: $84^\circ - 360^\circ = -276^\circ$ and $84^\circ - 2(360^\circ) = -636^\circ$ .
Example: $420^\circ$
- Nearest positive coterminals: $420^\circ - 360^\circ = 60^\circ$ and $420^\circ + 360^\circ = 780^\circ$ .
- Nearest negative coterminals: $420^\circ - 2(360^\circ) = -300^\circ$ and $420^\circ - 3(360^\circ) = -660^\circ$ .
Example: $-\frac{\pi}{3}$
- Nearest positive coterminals: $-\frac{\pi}{3} + 2\pi = \frac{5\pi}{3}$ and $\frac{5\pi}{3} + 2\pi = \frac{11\pi}{3}$ .
- Nearest negative coterminals: $-\frac{\pi}{3} - 2(2\pi) = -\frac{13\pi}{3}$ and $-\frac{\pi}{3} - 3(2\pi) = -\frac{19\pi}{3}$ .
Example: $\frac{5\pi}{2}$
- Nearest positive coterminals: $\frac{5\pi}{2} - 2π = \frac{\pi}{2}$ and $\frac{5\pi}{2} + 2π = \frac{9\pi}{2}$ .
- Nearest negative coterminals: $\frac{5\pi}{2} - 2(2π) = -\frac{3π}{2}$ and $\frac{5\pi}{2} - 3(2π) = -\frac{7π}{2}$ .

Definition: For a central angle $\theta$ measured in radians, the relationship is defined as $\theta = \frac{s}{r}$ , where $s$ is the intercepted arc length and $r$ is the radius.
The Arc Length Formula: $s = r\theta$ .
Mandatory Requirement: The angle $\theta$ MUST be in radians for this formula to function correctly.

Geography Case Study

The northeastern corner of Maine is due north of the southern tip of Chile.
Difference in latitude: $103^\circ$ .
Earth north-south circumference: $24,860 \text{ miles}$ .
Requirement: Calculate the distance between the two locations using both degree and radian measures.

Windshield Wiper Problem

A windshield wiper blade rotates through an angle of $135^\circ$ .
The bottom of the blade traces an arc with a $9 \text{-inch}$ radius.
The top of the blade traces an arc with a $23 \text{-inch}$ radius.
Conversion: Convert $135^\circ$ to radians: $135 \times \frac{\pi}{180} = \frac{3\pi}{4} \text{ radians}$ .
Top Arc Length Calculation: $s = 23 \times \left(\frac{3\pi}{4}\right) \approx 54.2 \text{ inches}$ .
Bottom Arc Length Calculation: $s = 9 \times \left(\frac{3\pi}{4}\right) \approx 21.2 \text{ inches}$ .
Result: The top arc is approximately $33 \text{ inches}$ longer ( $54.2 - 21.2 = 33$ ).

Convert $-495^\circ$ to radians: $-495 \times \frac{\pi}{180} = -\frac{11\pi}{4}$ .
Convert $\frac{13\pi}{12}$ radians to degrees: $\frac{13\pi}{12} \times \frac{180}{\pi} = 195^\circ$ .