Pre-Calculus
Notes: Section 4.1
Objectives:

Drawing Angles in Standard Position

Converting Angles (Radians and Degrees)

Using the Arc Length Formula

Finding Coterminal Angles

Converting angles that are in Degree, Minute, Second Form
Angles in Standard Position: An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the
terminal side.
Positive Angles are generated by a counter-clockwise rotation.
Negative Angles are generated by a clockwise rotation.
Practice Problems:
Sketch each angle in standard position.
1. 225

2.

170

3.

480


Radian Measure: One radian is defined as the central angle where the length of an arc and the radius of a circle are equal to each other.
Because of this relationship, the angle measure in radians can be thought of as the ratio between the arc length of a circle and its radius.
  r s
Furthermore, to find the angle measure for 1 revolution around the circle in radians, we can use the circumference formula ( C

2
 r ) to represent the arc length.
So we end up with the following relationship:
1 revolution around the circle:
  r s

2
 r r

2

.
The graph to the right shows some of the common radian measures that you will need to be familiar with.
Practice Problems for Sketching Angles in Radian Measure:
S ketch each angle In standard position.
1.
5

6
2.

7

4
3.
13

5

Converting Angles:
Since once around a circle is 360
 and 2
 radians, we can use the ratio 360
 convert angles from degrees to radians or radians to degrees.
Examples:
Convert 205
 into radians.
5

Convert
12
: 2
 into degrees.
or 180

:

to
Practice Problems:
1.
Convert

115
 into radians. 2. Convert
13

6
into degrees.
Coterminal Angles: Angles that have the same initial and terminal sides are coterminal angles.
To find coterminal angles in degrees:
Practice Problems:
To find coterminal angles in radians:
Determine two coterminal angles (one positive and one negative) for each angle.
1.
3

4
2.

120

3.

520


Degree, Minute, Second Form:
Example: thousandth.
Convert 54

1 5

3 2

into decimal form. Round the answer to the nearest
Practice Problems:
1. 45

3 5

2 0

2. 35

1 2

Using the Arc Length Formula:
The radian measure formula,
  s can be used to measure arc length along a circle. r
Specifically, for a circle of radius r, a central angle
 intercepts an arc of length s given by s
 r
 
Where

is measured in radians.
Example: Given: r = 9 cm,
 
130
 

180

13

180 s
 r
  
( 9 )
13


20 .
42 cm
18
Practice Problems:
Find the indicated variable:
1.
Find s
The length of arc AB is 20.42 cm.
2.

Find

In radians