Use your notes from last week:
• Find the value of x and y.
A)
1
y
45°
x
Use your notes from last week:
• Find the value of x and y.
B)
1
30°
x
y
Use your notes from last week:
• Find the value of x and y.
c)
1
60°
x
y
Section 7-1 Measurement of
Angles
Objective: To find the measure of an
angle in either degrees or radians and
to find coterminal angles.
What we are going to learn in
Sec 7.1
•
•
•
•
•
Vocabulary
Angle measure in degrees and radians
Standard position
The critical values on the Unit Circle
Coterminal angles
Section 7-1 Measurement of
Angles
• Objective:
1. To find the measure of an angle in either
degrees or radians.
2. To find coterminal angles.
Common Terms
• Initial ray is the ray that an angle starts
from.
• Terminal ray is the ray that an angle ends
on.
• Vertex
Common Terms
• A revolution is one complete circular
motion.
Angles in standard position
y
x
Standard Position
Section 4.1, Figure 4.2, Standard
Position of an Angle, pg. 248
Vertex at origin
Copyright © Houghton Mifflin Company. All rights reserved.
The initial
side
Digital Figures,
4–3 of an angle
in standard position is always
located on the positive x-axis.
Angles in standard position
•
•
•
•
The vertex of the angle is on (0,0).
Initial ray starts on the positive x-axis
The angle is measure counter clockwise.
The terminal ray can be in any of the
quadrants.
Angle describes the amount and direction of rotation
120°
–210°
Positive Angle: rotates counter-clockwise (CCW)
Negative Angle: rotates clockwise (CW)
PositiveSection
and negative
angles4.3, Positive and
4.1, Figure
Negative Angles, pg. 248
When sketching angles,
always use an arrow to
show direction.
Copyright © Houghton Mifflin Company. All rights reserved.
13
Digital Figures, 4–4
Units of angle measurement
•
There are two ways to measure an angle:
Degrees
&
Radians
Units of angle measurement
• There are two ways to measure an angle:
Degree: 1/360th of a circle. That is the
measure one sees on a protractor and
most people are familiar with.
Angles can be further split into 60
minutes per degree and 60 seconds per
minute.
Quadrantal Angle
• If the terminal ray of an angle in standard
position lies along an axis the angle is
called a quadrantal angle.
• The measure of a quadrantal angle is
𝜋
always a multiple of 90°, or
2
Quadrantal angles
Standard Position
• When an angle is shown in a
coordinate plane, it usually
appears in standard position,
with its vertex at the origin and its
initial ray along the positive xaxis.
Degrees
On one of the circles provided measure 1°
Radian Measure
• Use the string provided to
measure the radius.
• Start on the “x-axis” and use the
string to measure an arc the
same length on the circle.
• The angle created is one radian.
Angle θ is
one radian
Units of angle measurement
Radian: when the arc of
circle has the same
length as the radius of
the circle. Angle a
measures 1 radian.
Arc Lenght=radius
1
radian
Radius
approximations
• 1 radian ~ 57.2958 degrees
• 1 degree ~ 0.0174533 radians*
• *note: the radian measure is usually stated
as a fraction of .
Sec 7.1 day 2
Warm up
• While I check your UC, work on the following:
a) Display the measure of one radian on circle.
Display the measure of two radians on a cirlce.
b) Describe what one radian is in terms of the
radius of a circle r.
c) Draw a circle and identify a central angle.
Describe relationship between central angle
and the intercepting arc.
Find the measure of the central angle
𝟏
𝟐
The central angle shown has a measure of radian.
What is the length of arc 𝑪𝑫?
Find the measure of the central
angle COH
Length CGH = 4 cm
Measure of central angle:
• For radian measure:
s

r
s= arc length
r= radius
For degree measure:

180 s

r
Section 4.1, Figure 4.7, Common
Radian Measure
Radian Angles, pg. 249
30
Working with Radians
1 


180
180°
1 𝑟𝑎𝑑𝑖𝑎𝑛 =
𝜋
The conversion process
• When converting between the two units of
angle measurement, start with the
following template:
𝜋
𝑟𝑎𝑑𝑖𝑎𝑛
=
180° 𝐷𝑒𝑔𝑟𝑒𝑒
Example 1
Convert 196° to radians
𝜋
𝑟𝑎𝑑𝑖𝑎𝑛
=
180° 𝐷𝑒𝑔𝑟𝑒𝑒
𝜋
𝑟𝑎𝑑𝑖𝑎𝑛
=
180°
196°
Radian measure
196
is= 𝜋
180
=
49
𝜋
45
Convert
2
𝜋
3
to degrees.
𝜋
𝑟𝑎𝑑𝑖𝑎𝑛
=
180° 𝐷𝑒𝑔𝑟𝑒𝑒
2
𝜋
𝜋
3
=
180°
𝑥
2
180° × 𝜋
3
= 120°
𝑥=
𝜋
Calculator
• 2nd APP (Angle)
• Use DMS to convert to Degree, minute and
second.
• Use Angle to change 40° 20’ to a decimal
value.
• For more information click here
Coterminal angles
• Two angles in standard position are called
coterminal angles if they have the same
terminal ray.
• For any given angle there infinitely many
coterminal angles.
Example
• Find two angles, one positive and one
negative, that are coterminal with the angle
52°. Sketch all three angles
Solution
• 52 + 360 = 412
• 52 + 360 × 2 = 772
• 52 + 360 × 3 = 1132
• 52 − 360 = −308
• 52 − 360 × 2 = −668
• 52 − 360 × 3 = −1028
Example
• Find two angles, one positive and one
negative, that are coterminal with the
angle

4
•Sketch all three angles.
y
Coterminal Angles generalized:
• Degree measure: θ  360°n
• Radian measure: θ  2π n
• Where n is a counting number.
Helpful websites
• Trig flash cards
• http://mathmistakes.info/facts/TrigFacts/
• Hot math flash cards:
• http://hotmath.com/learning_activities/inter
activities/trig_flashcard.swf
Homework
• Sec 7.1 Written Exercises
• Problems # 1-8 all and
• # 9-29 odds
• UC with coordinates filled out.
1 degree = 60 minutes
1 minute = 60 seconds
1° = 60 
1  = 60 
3600
So … 1 degree = _________seconds
Express 365010as decimal degrees
36
50

60
10

3600
36 + .8333 + .00277
 36.8361
OR
Use your calculator!!
Express 365010as
decimal degrees
Enter 36
Press this button  ’ ’’
Press enter
Enter 50
Press this button  ’ ’’
Go over to the ’
symbol -- enter
Enter 10
Press this button  ’ ’’
Go over to the ’’
symbol -- enter
Press enter
Convert 50 47’ 50’’ to decimal degree
50.7972
Convert 125 27’ 6’’ to decimal degree
125.4517
Can you go backwards and convert the decimal degree to
degrees minutes seconds?
Enter 125.4517 Go to DMS hit enter.
Express 50.525 in degrees, minutes, seconds
50º + .525(60) 
50º + 36.5
50º + 36 + .5(60) 
50 degrees, 36 minutes, 30 seconds