Chapter 5: Trigonometric Functions – Angles, Arc Length, and Circular Motion

Trigonometric Functions

Section 5.1: 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 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 from which rotation starts is called the initial side, and the ray where rotation ends is the terminal side. Angles can be measured in degrees or radians, and their direction (counterclockwise or clockwise) determines whether they are positive or negative.

• Positive Angle: Formed by counterclockwise rotation from the initial side to the terminal side.

• Negative Angle: Formed by clockwise rotation from the initial side to the terminal side.

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

• Quadrantal Angle: An angle whose terminal side lies on the x-axis or y-axis.

Degree Measure

The degree is a common unit for measuring angles. One complete revolution is 360°, a right angle is 90°, and a straight angle is 180°.

• 1 revolution: $360^\circ$

• Right angle: $90^\circ$

• Straight angle: $180^\circ$

Drawing Angles in Standard Position

Angles can be drawn in standard position to visualize their terminal side. For example, 135° lies in quadrant II, −180° is a straight angle in the clockwise direction, 90° is a right angle, and 495° has the same terminal side as 135° due to periodicity.

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

Angles can be expressed in degrees, minutes, and seconds. One minute (1') is 1/60 of a degree, and one second (1") is 1/60 of a minute or 1/3600 of a degree.

• Conversion: To convert from degrees, minutes, seconds to decimal degrees, use: $\text{Decimal Degrees} = \text{Degrees} + \frac{\text{Minutes}}{60} + \frac{\text{Seconds}}{3600}$

• Example: $30^\circ 40' 10" = 30 + \frac{40}{60} + \frac{10}{3600} = 30.6694^\circ$ (rounded to four decimal places)

Radians

The radian is another unit for measuring angles. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

• Definition: $1$radian$= \frac{\text{Arc Length}}{\text{Radius}}$

• Relationship: $360^\circ = 2\pi$ radians

Find the Length of an Arc of a Circle

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

• Arc Length Formula: $s = r\theta$

• Example: For $r = 4$ meters and $\theta = 0.75$ radians, $s = 4 \times 0.75 = 3$ meters.

Convert from Degrees to Radians and from Radians to Degrees

Conversion between degrees and radians is essential in trigonometry.

• Degrees to Radians: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

• Radians to Degrees: $\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

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

Find the Area of a Sector of a Circle

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

• Sector Area Formula: $A = \frac{1}{2} r^2 \theta$

• Example: For $r = 3$ meters and $\theta = \frac{\pi}{4}$ radians (45°), $A = \frac{1}{2} \times 3^2 \times \frac{\pi}{4} \approx 3.53$ square meters.

Find the Linear Speed of an Object Traveling in Circular Motion

When an object moves along a circle, its linear speed and angular speed are related to the distance traveled and the angle swept out.

• Linear Speed (v): $v = \frac{s}{t}$, where s is the arc length and t is time.

• Angular Speed (\omega): $\omega = \frac{\theta}{t}$, where θ is the angle in radians.

• Relationship: $v = r\omega$

• Example: A rock spun at the end of a 2-foot rope at 180 revolutions per minute has a linear speed calculated using these formulas.

Applications: Distance Between Two Cities

Trigonometric concepts can be applied to real-world problems, such as finding the distance between two cities using their latitudes and the radius of the Earth.

• Central Angle: The angle between two locations on Earth is calculated using their latitudes.

• Distance Formula: $s = r\theta$, where r is Earth's radius and θ is the central angle in radians.

• Example: Sioux Falls and Dallas are separated by a central angle, and their distance is found using the arc length formula.

Angle Type	Definition	Example
Positive Angle	Counterclockwise rotation	135°
Negative Angle	Clockwise rotation	−180°
Quadrantal Angle	Terminal side on x- or y-axis	0°, 90°, 180°, 270°

Unit	Conversion
Degrees to Radians	$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$
Radians to Degrees	$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

Additional info: The notes have been expanded to include full definitions, formulas, and examples for each concept, ensuring completeness and clarity for Precalculus students.