Trigonometric Functions

Section 5.1: Angles, Arc Length, and Circular Motion

This section introduces the foundational concepts of angles, their measurement, and their applications in circular motion. Understanding these concepts is essential for the study of trigonometric functions and their real-world applications.

Angles and Degree Measure

An angle is formed by two rays sharing a common endpoint called the vertex. The ray from which the angle starts is the initial side, and the ray where the angle ends is the terminal side. The direction of rotation determines the sign of the angle: counterclockwise is positive, and clockwise is negative.

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

Degrees

• One full revolution (counterclockwise) is 360°.

• A right angle is 90° (one-quarter revolution).

• A straight angle is 180° (half revolution).

Examples: Drawing Angles in Standard Position

• 135°: Lies in the second quadrant.

• −180°: Half revolution in the clockwise direction.

• 90°: One-quarter revolution counterclockwise.

• 495°: Equivalent to 135° (495° − 360° = 135°).

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

Angles can be measured in degrees, minutes, and seconds (DMS):

• 1 degree (1°) = 60 minutes (60')

• 1 minute (1') = 60 seconds (60")

• Example: 30° 40' 10"

To convert between decimal degrees and DMS:

• Decimal to DMS: Multiply the decimal part by 60 to get minutes, then multiply the decimal part of minutes by 60 to get seconds.

• DMS to Decimal: Degrees + (minutes/60) + (seconds/3600)

Radians

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

• For a circle of radius r, an angle of 1 radian subtends an arc of length r.

• There are radians in a full circle (360°).

Find the Length of an Arc of a Circle

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

Example: For a circle of radius 4 meters and a central angle of 0.75 radians:

meters

Convert from Degrees to Radians and from Radians to Degrees

To convert between degrees and radians:

• Degrees to radians:

• Radians to degrees:

Common Angles in Degrees and Radians

Find the Area of a Sector of a Circle

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

Example: For a sector with radius 3 meters and angle 45° ( radians):

square meters

Find the Linear Speed of an Object Traveling in Circular Motion

When an object moves along a circle of radius r at a constant speed, the linear speed v is:

where s is the arc length traveled in time t.

Angular Speed

The angular speed is the angle (in radians) swept out per unit time:

where is in radians and is time.

Example: Linear Speed in Circular Motion

A child spins a rock at the end of a 2-foot rope at 180 revolutions per minute. The linear speed is found by first calculating the angular speed and then using the formula for linear speed.

Applications: Distance Between Two Cities

The distance between two points on a circle (such as cities on Earth) can be found using the arc length formula, where the central angle is the difference in latitude (converted to radians) and the radius is the radius of Earth.