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?)

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