Angles, Arc Length, and Circular Motion: Precalculus Study Notes

Angles and Degree Measure

Definition of an Angle

An angle is formed by rotating a ray (called the initial side) about its endpoint (the vertex) to a new position (the terminal side). Angles are fundamental in trigonometry and are measured in degrees or radians.

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

• Initial Side: The starting position of the ray.

• Terminal Side: The position of the ray after rotation.

• Vertex: The common endpoint of the initial and terminal sides.

Angles are measured in a counterclockwise direction from the initial side to the terminal side. Positive angles are measured counterclockwise, and negative angles are measured clockwise.

Standard Position and Quadrants

• 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 of an angle in standard position may lie in any of the four quadrants or along the axes (quadrantal angles).

Quadrants:

• Quadrant I: 0° to 90°

• Quadrant II: 90° to 180°

• Quadrant III: 180° to 270°

• Quadrant IV: 270° to 360°

Degree Measure

The degree is a unit of angle measure. One complete revolution is 360 degrees.

• 1 revolution = 360°

• Right angle = 90°

• Straight angle = 180°

Example: Drawing Angles

• 45°: Terminal side in Quadrant I

• -90°: Terminal side on negative y-axis (clockwise rotation)

• 225°: Terminal side in Quadrant III

• 405°: Terminal side in Quadrant I (one full revolution plus 45°)

Convert between Decimal and Degree, Minute, Second Measures for Angles

Degree, Minute, Second (DMS) Notation

Angles can be expressed in degrees (°), minutes ('), and seconds ('').

• 1 degree (1°) = 60 minutes (60')

• 1 minute (1') = 60 seconds (60'')

To convert between decimal degrees and DMS:

• Decimal to DMS: Separate the whole number (degrees), multiply the decimal by 60 to get minutes, and multiply the remaining decimal by 60 to get seconds.

• DMS to Decimal: $\text{Decimal Degrees} = \text{Degrees} + \frac{\text{Minutes}}{60} + \frac{\text{Seconds}}{3600}$

Example: Conversion

• Convert 50° 6' 21'' to decimal degrees:

• $50 + \frac{6}{60} + \frac{21}{3600} = 50.1058°$ (rounded to four decimal places)

• Convert 21.256° to DMS:

• 21°; 0.256 × 60 = 15.36 → 15'; 0.36 × 60 ≈ 22'' → 21° 15' 22''

Radian Measure

Definition of a Radian

A radian is the measure of a central angle that intercepts an arc equal in length to the radius of the circle. Radians are the standard unit of angular measure in mathematics.

• 1 radian = angle subtended by an arc equal to the radius

• There are $2\pi$ radians in one complete revolution

Find the Length of an Arc of a Circle

Arc Length Formula

The length of an arc (s) of a circle with radius r, subtended by a central angle $\theta$ (in radians), is given by:

• $s = r\theta$

Note: The angle $\theta$ must be in radians for this formula to be valid.

Example: Arc Length

• Find the length of the arc of a circle of radius 2 meters subtended by a central angle of 0.25 radians:

• $s = 2 \times 0.25 = 0.5$ meters

Convert from Degrees to Radians and from Radians to Degrees

Conversion Formulas

• 1 revolution = $2\pi$ radians

• $180^\circ = \pi$ radians

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

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

Example: Degrees to Radians

• Convert 60° to radians: $60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians

Example: Radians to Degrees

• Convert $\frac{\pi}{6}$ radians to degrees: $\frac{\pi}{6} \times \frac{180}{\pi} = 30^\circ$

Summary Table: Degree and Radian Conversions

Degrees	Radians
0°	0
90°	$\frac{\pi}{2}$
180°	$\pi$
270°	$\frac{3\pi}{2}$
360°	$2\pi$

Practice Problems

• Convert 150° to radians.

• Convert $\frac{3\pi}{2}$ radians to degrees.

• Find the arc length of a circle with radius 5 units and central angle 2 radians.

• Express 37.75° in degree, minute, second notation.

Additional info: These notes cover foundational concepts in trigonometry and circular motion, essential for Precalculus students. The examples and formulas provided are standard in college-level mathematics courses.