Circular Functions
Radian Measure
Text :
3.2 Radians and Degrees
3.4 Arc Length and Area of a Sector
3.5 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 Distance
Linear Distance (Arc Length)
Area of a Sector
Example 1 pg 119
A 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 Sector
A = ½ r²Ɵ
Applications
Angular Distance
Linear 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 #7
Angle 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