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

Trigonometric Functions

Section 7.1: Angles, Arc Length, and Circular Motion

This section introduces foundational concepts in trigonometry, including the measurement of angles, conversion between units, arc length, sector area, and the analysis of circular motion. These topics are essential for understanding trigonometric functions and their applications in precalculus.

Angles and Degree Measure

Angles are formed by two rays sharing a common vertex. The ray where measurement begins is called the initial side, and the ray where measurement ends is the terminal side. The direction of rotation determines the sign of the angle:

• Counterclockwise rotation: Positive angle

• Clockwise rotation: Negative angle

An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.

Angles in standard position can be classified as:

• Quadrantal angle: Terminal side lies on the x-axis or y-axis

• Quadrant angle: Terminal side lies in a quadrant (I, II, III, IV)

Common angle measures:

• One degree (1°): 1/360 of a revolution

• Right angle: 90°, or 1/4 revolution

• Straight angle: 180°, or 1/2 revolution

Degree, Minute, Second Notation

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

• 1° = 60'

• 1' = 60"

Example: Convert 65º9'17" to decimal degrees:

• Decimal degrees = 65 + 9/60 + 17/3600 = 65.1547° (rounded to four decimal places)

Example: Convert 32.479º to degree, minute, second notation:

• 32º

• 0.479 × 60 = 28.74 → 28'

• 0.74 × 60 = 44.4 → 44"

• Result: 32º28'44"

Arc Length

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

$s = r\theta$

Example: Find the arc length for r = 4 meters, θ = 0.75 radians:

• $s = 4 \times 0.75 = 3$ meters

Radians and Degrees

Angles can be measured in degrees or radians. The conversion formulas are:

• $1 \text{ revolution} = 2\pi \text{ radians}$

• $180^\circ = \pi \text{ radians}$

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

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

Example: Convert 45° to radians:

• $45^\circ \times \frac{\pi}{180} = \frac{\pi}{4}$ radians

Example: Convert $\frac{2\pi}{3}$ radians to degrees:

• $\frac{2\pi}{3} \times \frac{180}{\pi} = 120^\circ$

Applications: Field Width of a Camera Lens

For small angles, the arc length approximates the chord length. The field width of a camera lens can be estimated using the arc length formula, with the viewing angle converted to radians.

Example: For a 400mm lens at 725 feet, viewing angle 6º9′:

• Convert 6º9′ to radians

• Use $s = r\theta$ with r = 725 feet

Area of a Sector

The area (A) of a sector formed by a central angle (θ, in radians) in a circle of radius r is:

$A = \frac{1}{2} r^2 \theta$

Example: Find the area for r = 3 meters, θ = 45° ($\frac{\pi}{4}$ radians):

• $A = \frac{1}{2} \times 3^2 \times \frac{\pi}{4} = 3.53$ square meters (rounded)

Linear Speed

Linear speed (v) is the distance traveled per unit time along a circle:

$v = \frac{s}{t}$

Where s is arc length and t is time.

Angular Speed

Angular speed (\omega) is the angle swept out per unit time:

$\omega = \frac{\theta}{t}$

Where θ is in radians and t is time.

Example: Finding Linear Speed

A child spins a rock at the end of a 2-foot rope at 180 revolutions per minute (rpm). Find the linear speed when released:

• Radius r = 2 feet

• Angular speed $\omega = 180 \times 2\pi = 360\pi$ radians/min

• Linear speed $v = r\omega = 2 \times 360\pi = 720\pi$ ft/min ≈ 2262 ft/min ≈ 25.7 mi/h

Additional info: All formulas and examples are expanded for clarity and completeness. Images are included only when directly relevant to the explanation.