Chapter 4 Section 1: Measures of Angles and Rotations

Angle Definition: In geometry, an angle is defined as two rays that share the same endpoint.
Vertex: This shared endpoint is typically referred to as the vertex of the angle.
Standard Position: An angle is said to be in standard position when the following two conditions are met: - Its Initial Side is located on the positive $x$ -axis. - Its Vertex is located at the origin ( $0, 0$ ) of the coordinate plane.

Angle measures can be determined by the direction of rotation from the initial side to the terminal side: - Counterclockwise Rotation: This direction results in Positive degree measures. - Clockwise Rotation: This direction results in Negative degree measures.
Example Degree Measures: - $45^{\circ}$ - $90^{\circ}$ - $180^{\circ}$ - $300^{\circ}$ - The transcript also notes other specific measures including $180$ , $300$ , $945$ , and $-1900$ .

There are three primary ways to measure an angle discussed in this section: - Degrees: The most common unit of angular measurement ( $360^{\circ}$ in a full circle). - Revolutions: - This represents the number or portion of a full circle that an angle makes. - Key characteristic: Revolution values themselves are not inherently positive or negative; instead, they include an explicit direction (Clockwise or Counterclockwise). - Radians: - This is a measurement system based upon the arc length created by the angle on a circle. - The arc length is a specific portion of the total circumference $C$ . - Circumference Formulas: - $C = 2\pi r$ (where $r$ is the radius). - $C = \pi d$ (where $d$ is the diameter).

Goal: To be able to convert between units and visualize angle sizes effectively.
Practice: Converting Degrees to Revolutions: - To find the revolution measure, divide the degrees by $360^{\circ}$ . - Case 1: $-240^{\circ}$ - Calculation: $\frac{240}{360} = \frac{2}{3}$ . - Result: $\frac{2}{3}$ revolutions clockwise (negative sign indicates clockwise direction). - Case 2: $30^{\circ}$ - Calculation: $\frac{30}{360} = \frac{1}{12}$ . - Result: $\frac{1}{12}$ revolutions counterclockwise (positive sign indicates counterclockwise direction).
Practice: Converting Revolutions to Degrees: - Multiply the revolution fraction by $360^{\circ}$ . - Case 1: $\frac{5}{6}$ of a revolution counterclockwise. - Calculation: $\frac{5}{6} \times 360^{\circ} = 300^{\circ}$ . - Case 2: $\frac{1}{3}$ of a revolution clockwise. - Calculation: $\frac{1}{3} \times 360^{\circ} = 120^{\circ}$ . Since it is clockwise, the final measure is given as $-120^{\circ}$ .

Radian: This is the angle measure that is exactly equal to the length of the arc that the angle creates on a unit circle.
The Unit Circle: - A unit circle is a circle with a radius of exactly $1$ unit ( $r = 1$ ). - Because different angles create different arc lengths on this circle, we can measure angles by measuring these specific arc lengths.
Why does $360^{\circ} = 2\pi$ Radians?: - On a unit circle, the radius $r = 1$ . - The formula for the circumference is $C = 2\pi r$ . - Substituting the radius of the unit circle: $C = 2\pi(1) = 2\pi$ . - Therefore, one full trip around the circle ( $360^{\circ}$ ) is equivalent to $2\pi$ radians.

Establishing these common measurements helps in rapid visualization: - $360^{\circ}$ : $2\pi$ radians. - $180^{\circ}$ : $\pi$ radians. - $90^{\circ}$ : $\frac{\pi}{2}$ radians. - $45^{\circ}$ : $\frac{\pi}{4}$ radians. - $130^{\circ}$ : To convert, use the formula $130^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{13\pi}{18}$ radians.

To convert from degrees to radians, multiply the degree measure by $\frac{\pi}{180^{\circ}}$ .
Practice Problems: - $210^{\circ}$ : - Calculation: $210 \times \frac{\pi}{180} = \frac{21\pi}{18} = \frac{7\pi}{6}$ . - $-150^{\circ}$ : - Calculation: $-150 \times \frac{\pi}{180} = -\frac{15\pi}{18} = -\frac{5\pi}{6}$ . - $570^{\circ}$ : - Calculation: $570 \times \frac{\pi}{180} = \frac{57\pi}{18} = \frac{19\pi}{6}$ .

Key Facts to Memorize: - $1$ full revolution = $360^{\circ} = 2\pi$ radians. - $\frac{1}{2}$ revolution = $180^{\circ} = \pi$ radians.
Important Rule: Revolutions cannot be expressed as negative numbers; the direction must be stated as "clockwise" or "counterclockwise" instead.
Methods: Proportions or dimensional analysis are reliable ways to convert between these units.

Homework: Complete problems $1$ through $4$ from the Section $4.1$ assignment.