Download Section 7-1 Measurement of Angles

Warm Up
1)Find the inverse in expanded form:
f x = −4 +
𝑥−5
8
2
𝑥
2
𝑦
+
=7
2
2
5𝑥 − 𝑦 = 1
2
3) Factor: 5𝑥 + 3𝑥 − 8
2
4) Simplify: 5 2𝑥 − 3
2)Solve the system:
Final Exam
Average
Median
nd
2
79.3
81
rd
3
85.8
85.8
th
4
82.5
88
Chapter 7
Trigonometric Functions
Section 7-1 Measurement of Angles
Objective: To find the measure of
an angle in either degrees or radians
and to find coterminal angles.
Common Terms
• Initial ray - the ray that an angle starts from.
• Terminal ray - the ray that an angle ends on.
• Vertex – the starting point
• A revolution is one complete circular motion.
Standard Position of an Angle
• The vertex of the angle is at (0,0).
• Initial ray starts
on the positive x-axis.
Section 4.1, Figure 4.2, Standard
Position
of anbe
Angle,
pg. 248
• The terminal
ray can
in any
of the quadrants.
The vertex is at origin
The initial side is located
on the positive x-axis
The angle describes the amount and direction of rotation.
120°
When sketching
angles, always
use an arrow to
show direction.
–210°
Positive Angle: rotates counter-clockwise (CCW)
Negative Angle: rotates clockwise (CW)
Units of Angle Measurement
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.
25 20 '6"
25 degrees, 20 minutes, and 6 seconds
Degrees, Minutes, & Seconds to a Decimal Approximation
25 20 '6"
Divide the
minutes by 60
Divide the
seconds by 3600
 20   6 
 25     

 60   3600 
 25.335
Decimal Approximation to Degrees & Minutes
12.3
Multiply the
tenths by 60
 12  0.3(60) '
 12 18'
Units of Angle Measurement
Radian
• 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
Arc Length = Radius
When the arc of circle has the same length as
the radius of the circle, angle  measures 1 radian.
Units of Angle Measurement
Radian
• Use the string provided to show an
angle of 2 radians.
• How many radians make a complete
circle?
Units of Angle Measurement
Radian
• Use the string provided to show an
angle of 2 radians.
• How many radians make a complete
circle?
Find the measure of the central angle in
Radians.
2 Radians
𝟏
𝟐
The central angle shown has a measure of radian.
What is the length of arc 𝑪𝑫?
2 inches
Find the measure of the central angle COH
Length CGH = 4 cm
4
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
5
Measure of Central Angle
For radian measure:
For degree measure:
s

r

180 s

r
s= arc length
r= radius
1 revolution =
360 degrees = 2 radians
Fill in each unit circle with the degree
and radian measure for each line.
360  2 radians
Conversion Formulas:
180   radians
To convert degrees to radians, multiply by
𝝅
𝟏𝟖𝟎
To convert radians to degrees, multiply by
𝟏𝟖𝟎
𝝅
Convert 196˚ to radians.
196˚∗
𝜋
180˚
Convert 1.35 radians to degrees.
=
196𝜋
180
=
49𝜋
45
radians
180˚
1.35 ∗
= 77.35˚
𝜋
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.
Find two angles, one positive and one negative, that are
coterminal with the angle 52°. Sketch all three angles.
• 52˚ + 360˚ = 412˚
• 52˚ + 360˚ × 2 = 772˚
• 52˚ + 360˚ × 3 = 1132˚
52˚
-308˚
• 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.
Find two angles, one positive and one
𝜋
negative, that are coterminal with the angle
•
𝜋
4
•
𝜋
4
•
𝜋
4
+ 2𝜋 =
9𝜋
4
+ 4𝜋 =
17𝜋
4
+ 6𝜋 =
25𝜋
4
4
•
𝜋
4
•
𝜋
4
•
𝜋
4
− 2𝜋 =
7𝜋
−
4
− 4𝜋 =
15𝜋
−
4
− 6𝜋 =
23𝜋
−
4
Coterminal Angles Generalized:
• Degree measure: θ  360°n
• Radian measure: θ  2π n
where n is a counting number
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 always a multiple of
𝜋
90° or
2
Homework
Page 261 #1-31 odds