Precalc
Mrs. Caruso
Guided Notes for Section 4.1
Angles and their Measures
Vocabulary
Ray:
Angle:
Initial Side: (picture to show sides)
Terminal Side:
Vertex:
Methods to Name an Angle
Often we use Greek letters: Alpha α, Beta β, Gamma γ, and Theta θ
 Recall from Geometry, you can name angles with three letters with the vertex
letter in the middle.
Standard Position: An angle is in standard position if:
 Its vertex is at the origin of a rectangular coordinate system
 Its initial side lies along the positive x-axis
Example of angles in standard position
Two types of rotation
 Counterclockwise rotation
 Clockwise rotation
Positive Angles are created by counterclockwise rotation.
Negative Angles are created by clockwise rotation.
Pictures
When an angle is in standard position, we say that the angle lies in that quadrant.
Quadrantal Angle: an angle whose terminal side lies on the x-axis or y-axis. All angles
do not have to lie in a quadrant.
Examples of quadrantal angles
Measuring Angles Using Degrees
Angles are measured by determining the amount of
How many degrees are in a circle?
Think of a clock: It is now 12 pm. How many degrees will it take to get back to the 12?
Quick Review from Geometry
Right Angle:
Acute Angle:
Obtuse Angle:
Straight Angle:
We will often use Theta (θ) to represent the measure of an angle. For example, θ = 45
Drawing Angles in Standard Position
Examples:
1) 45
2) 225
3) 135
4) 405
Try These!
1) 30
2) 210
3)  120
4) 390
Coterminal Angles
Coterminal Angles: two angles with the same initial and terminal sides
Examples: 45 and 405 , 225 and 135
Why do we use Standard Position for Angles?
Every angle has
.
One or more complete
results in
.
Coterminal Angles Definition
An angle of x is coterminal with angles of x  k  360 , where k is an integer.
How to find Coterminal Angles?
Add or Subtract 360 from the angle in standard position.
Examples:
Assume the following angles are in standard position. Find a positive angle less then
360 that is coterminal with:
1) a 420 angle
2) a 120 angle
Try These!
1) a 400 angle
2) a 135 angle
Measuring Angles Using Radians
Definition of a One Radian: one radian is a measure of the central angle of a circle that
intercepts an arc equal in length to the radius of the circle.
Visualize This!
The radian measure of any central angle is
.
Definition of Radian Measure:
Examples of Computing Radian Measure:
1) A central angle, θ, in a circle of radius 6 inches intercepts an arc of length 15
inches. What is the radian measure of θ?
2) A central angle, θ, in a circle of radius 12 feet intercepts an arc of length 42 feet.
What is the radian measure of θ?
Try This!
3) A central angle, θ, in a circle of radius 10 inches intercepts an arc of length 40
inches. What is the radian measure of θ?
The Relationship between Radians and Degrees
360 rotates a ray back onto itself. How many radians is that?
What is the length of intercepted arc equal to?
Converting between Degrees and Radians
Using the basic relationship
.
1) To convert degrees to radians,
2) To convert radians to degrees,
There should never be any confusion about what
unit you are using.
ALWAYS LABEL YOUR DEGREES.
Often, you will see radians written for that measurement, but sometimes no unit will
be written. We then assume you are using radians.
Examples:
Convert each angle in degrees to radians:
1) 30
2) 90
3) 135
Convert each angle in radians to degrees:
1)

radians
3
2)
 5
radians
3
3) 1 radian
Try These!
Convert each angle in degrees to radians:
1) 60
3)  300
2) 270
Convert each angle in radians to degrees:
1)

radians
4
2)
 4
radians
3
3) 6 radians
The Length of a Circular Arc
Finding the length of an arc of a circle.
Picture:
Examples:
1) A circle has a radius of 10 inches. Find the length of the arc intercepted by a
central angle of 120 .
2) A circle has a radius of 6 inches. Find the length of the arc intercepted by a central
angle of 45 . Express arc length in terms of  . Then round your answer to two
decimal places.