Angles, Arc Length, and Circular Motion: Study Notes for Precalculus (Trigonometric Functions)

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 concepts are essential for understanding trigonometric functions and their applications in precalculus.

Angles and Degree Measure

An angle is formed by two rays with a common vertex. The ray where the rotation starts is called the initial side, and the ray where the rotation ends is the terminal side. The direction of rotation determines the sign of the angle:

• Counterclockwise rotation: Positive angle

• Clockwise rotation: Negative angle

Standard Position of Angles

An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x-axis. The terminal side determines whether the angle is positive or negative.

Quadrants and Quadrantal Angles

When an angle is in standard position, its terminal side may lie in a quadrant or on an axis:

• If the terminal side lies in a quadrant, the angle is said to lie in that quadrant.

• If the terminal side lies on the x-axis or y-axis, the angle is a quadrantal angle.

Degree, Right Angle, and Straight Angle

Angle measures are commonly given in degrees:

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

• Right angle: 90°, or 1/4 revolution

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

Drawing Angles in Standard Position

To draw angles in standard position, start at the positive x-axis and rotate according to the angle's measure and direction. For example:

• 135°: 1.5 right angles, lies in quadrant II

• –180°: 0.5 revolution clockwise, terminal side on negative x-axis

• 90°: 0.25 revolution counterclockwise, terminal side on positive y-axis

• 495°: 1 full revolution (360°) plus 135°, terminal side same as 135°

Minutes and Seconds

Angles can be measured in degrees, minutes, and seconds:

• 1 revolution = 360°

• 1° = 60' (minutes)

• 1' = 60" (seconds)

Converting Between Decimal Degrees and Degree-Minute-Second Notation

To convert from degree-minute-second (DMS) to decimal degrees:

• Decimal degrees = degrees + (minutes/60) + (seconds/3600)

To convert from decimal degrees to DMS:

• Minutes = decimal part × 60

• Seconds = decimal part of minutes × 60

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

• 65 + 9/60 + 17/3600 ≈ 65.1547°

Example: Convert 32.479º to DMS:

• 32º

• 0.479 × 60 = 28.74 → 28'

• 0.74 × 60 ≈ 44"

• Result: 32º28'44"

Arc Length

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

• $s = r\theta$

For a full revolution (θ = 2π radians):

• $s = 2\pi r$

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

• $s = 4 \times 0.75 = 3$ meters

Radians and Degrees

Relationship between radians and degrees:

• 1 revolution = $2\pi$ radians = 360°

• $180^ ext{o} = \pi$ radians

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

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

Converting Between Degrees and Radians

Example: Convert 45° to radians:

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

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

• $\frac{\pi}{3} \times \frac{180}{\pi} = 60^ ext{o}$

Applications: Field Width of a Camera Lens

For small angles, the arc length subtended by a central angle is approximately equal to the chord length. This is used to estimate the field width of a camera lens:

• Convert the viewing angle to radians

• Use arc length formula: $s = r\theta$

Area of a Sector

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

• $A = \frac{1}{2} r^2 \theta$

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

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

Linear Speed

Linear speed v is the distance traveled per unit time by an object moving along a circle of radius r:

• $v = \frac{s}{t}$

• Where s is arc length and t is time

Angular Speed

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

• $\omega = \frac{\theta}{t}$

• Where θ is the angle in radians and t is time

Linear speed and angular speed are related by:

• $v = r\omega$

Example: Finding Linear Speed

A child spins a rock at the end of a 2-foot rope at 180 revolutions per minute (rpm):

• Radius r = 2 feet

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

• Linear speed $v = r\omega = 2 \times 180 \times 2\pi = 2262$ ft/min

• Convert to miles/hour: $2262 \text{ ft/min} \approx 25.7 \text{ mi/h}$