Section 4.1 Radian and Degree Measure

Angles

Radian Measure

Definition of a Radian

One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.

Example 1: Finding Coterminal Angles

Find a positive and a negative angle coterminal with a  angle.

A  angle and a  angle are coterminal with a  angle.

Two positive angles are complementary (complements of each other) if their sum is π/2.  Two positive angles are supplementary (supplements of each other) if their sum is π. 

Degree Measure

Conversions Between Degrees and Radians

1. To convert degrees to radians, mulitply degrees by π rad/180°

2. To convert to degrees, multiply radians by 180°/π rad

Example 2: Converting from Degrees to Radians/ Radians to Degrees

a. 135°= (135 deg) (π rad/180 deg) = 3π/4

b. - π/2 rad = (-π/2 rad) (180 deg/π rad) = -90°

Applications

The radian measure formula  θ = s/r can be used to measure arc length along a circle. Specifically, for a circle, of radius r, a central angle  θ intercepts an arc of length s given by

where θ is measured in radians.

Example 3: Finding Arc Length

a. A circle has a radius of 4 inches.  Find the length of the arc intercepted by a central angle of 240°.

240° = (240 deg) (π rad/180 deg)

        = 4π/3 radians

       = 4(4π/3)

       = 16π/3

       = 16.76 inches

Linear and Angular Speed

Consider a particle moving at a constant speed along a circular arc of radius r.  If s is the length of the arc traveled in time t, then the linear speed of the particle is

               Linear Speed = Arc lenth/time = s / t

Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, the angular speed of the particle is

               Angular speed = central angle/time = θ / t

Example 4: Finding Linear Speed

Example 5: Finding Angular and Linear Speed