Chapter 4: Circular Functions
Chapter 4: Circular FunctionsLesson 1: Measures of Angles and Rotations
Mrs. ParzialeDo NowGiven a radius of 1 for the circle to the right, find the following in terms of pi ()The circumference of the circle.The length of a 180 arc.The length of a 90 arc.The length of a 45 arc.1Terms To Knowangle the union of two rays with a common endpoint.
sides are examples
vertex The point at which the two rays meet. B is the vertex in this example.
More Terms to Knowrotation image - is the rotation image of about the vertex B
counterclockwise rotations are positive.clockwise rotations are negative.
Measure of an angle represents its size and direction.RevolutionsRotations can be measured in revolutions.
1 counterclockwise revolution = 360
To convert To convert revolutions todegrees to degrees:revolutions:
Example 1:(a) revolution counterclockwise
(b) revolution clockwise(c) revolution clockwise
(d) revolutions counterclockwise
RadiansRadians have only been around for about 100 years.Radians are another means of measuring angle based on how far it travels on the unit circle.Primary use of radians was to simplify calculations using angle measures.Relate to the circumference of a circle with radius of 1
More Radianscircumference of a circle
circumference of a circle with radius of 1
With one revolution of a circle
Example 2: Convert to radians. Give exact values (in terms of pi):
revolution counterclockwise
revolution clockwise
revolution counterclockwise
revolution clockwise
1 rev = 2Example 2, cont: Convert to radians. Give exact values:
(e) revolution clockwise
(f) revolution counterclockwise
Unit CircleConverting radians to degrees:
Converting degrees to radians:
Unit Circle
How many radians in 30?
How many degrees in ?
Example 3: Convert to radians. Give both exact and approximate values (hundredth):
Example 4: How many revolutions equal 8 radians (approx)? (Set up a proportion.)
Example 5: How many revolutions equal radians (exact?)
beatles-revolution56397.42