Circular Functions

Radian Measure

Text : 3.2 Radians and Degrees3.4 Arc Length and Area of a Sector3.5 Velocities

Objectives

• To Define Radian Measure using central angles of rotation.

• Convert from Degree to Radian measurement.

• Applications of Radian Measure– Angular Distance– Linear Distance (Arc Length)– Angular and Linear Velocities

Warm-Up

1. Rotating Light The red light on the top of a police car rotates through one complete revolution every 2 seconds. Through how many degrees does it rotate in 1 second?

2. Rotating Light A searchlight rotates through one complete revolution every 4 seconds. How long does it take the light to rotate 90°?

3. Clock Through how many degrees does the hour hand of a clock move in 4 hours?

4. Rotation of the Earth It takes the earth 24 hours to make one complete revolution on its axis. Through how many degrees does the earth turn in 12 hours?

Additional Examples Pg 128 #95

The Campagnolo Hyperon carbon wheel has 22 spokes evenly distributed around the rim of the wheel. What is the measure, in radians, of the central angle formed by adjacent pairs of spokes?

Formulas

Angular DistanceLinear Distance (Arc Length)Area of a Sector

Example 1 pg 119A central angle Ɵ in a circle of radius 3 cm cuts off an arc length of 6 cm. What is the radian measure of Ɵ ?

Example 1 pg 141

Give the length of the arc cut off by the central angle of 2 radians in a circle of radius 4.2 inches.

The Area of a SectorA = ½ r²Ɵ

Applications

Angular DistanceLinear Distance (Arc

Length)

In navigation, distance is not usually measured along a straight line, but along a great circle because the Earth is round.

Additional Examples pg 125 #7Angle Between Cities Los Angeles and San Francisco are approximately 450 miles apart on the surface of the earth. Assuming that the radius of the earth is 4000 miles, find the radian measure of the central angle with its vertex at the center of the earth that has Los Angeles on one side and San Francisco on the other side.

Additional Example Pg 126 # 21 and 22

If a central angle with its vertex at the center of the earth has a measurement of 1‘, then the arc on the surface of the earth that is cut off by this angle (known as the great circle distance) has a measure of 1 nautical mile.

(Note: a ‘regular’ mile is a statute mile.)

Example 2 Pg 141

Below is a model of a Ferris wheel with diameter 250 ft, and Ɵ is a central angle formed as a rider travels from his initial position P₀ to position P₁. Find the distance traveled by the rider if Ɵ =45° and if Ɵ =105°.

Example 3 pg 141

Example 4 Pg 143

Velocities