Chapter 4: Circular Functions

Lesson 1: Measures of Angles and Rotations

Mrs. Parziale

Do Now

• Given a radius of 1 for the circle to the right, find the following in terms of pi ()

1. The circumference of the circle.2. The length of a 180° arc.

3. The length of a 90° arc.

4. The length of a 45° arc.1

Terms To Know

• angle – 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.

BA and BC����������������������������

CB

A

More Terms to Know

• rotation image - is the rotation image of

about the vertex B

• counterclockwise rotations – are positive.

• clockwise rotations – are negative.

CB

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BC��������������

Measure of an angle

represents its size

and direction.

Revolutions

• Rotations can be measured in revolutions.

• 1 counterclockwise revolution = 360°

To convert To convert revolutions to degrees to degrees: revolutions:(360 )rev

deg

360

Example 1:

(a) revolution counterclockwise

(b) revolution clockwise (c) revolution clockwise

(d) revolutions counterclockwise

1

31

62

131

38

Radians

• Radians have only been around for about 100 years.

• Radians are another means of measuring angle sides.

• Primary use of radians was to simplify calculations using angle measures.

• Relates to the circumference of a circle with radius of 1

More Radians

• circumference of a circle

• circumference of a circle with radius of 1

• With one revolution of a circle

2C r

2 2 (1) 2C r

2 360radians

Example 2: • Convert to radians. Give exact values (in terms of pi):

a) revolution counterclockwise

b) revolution clockwise

c) revolution counterclockwise

d) revolution clockwise

1

2

2

3

1

3

1

6

12

2

1 rev = 2

Example 2, cont:

• Convert to radians. Give exact values:

(e) revolution clockwise

(f) revolution counterclockwise

21

3

13

8

Unit Circle

• Converting radians to degrees:

• Converting degrees to radians:

deg180

radians

180deg radians

Unit Circle

deg180

radians

180deg radians

How many radians in 30°?

How many degrees in ?5

6

Example 3:

• Convert to radians. Give both exact and approximate values (hundredth):

. 60a . 165d . 90c . 45b

Example 4:

• How many revolutions equal 8 radians (approx)? (Set up a proportion.)

Example 5:

• How many revolutions equal radians (exact?)

5

7