Warm Up1)Find the inverse in expanded form:
f x = −4 +𝑥−5
8
2)Solve the system: 𝑥2 + 𝑦2 = 7
5𝑥2 − 𝑦2 = 1
3) Factor: 5𝑥2 + 3𝑥 − 8
4) Simplify: 5 2𝑥 − 3 2
1 revolution =
360 degrees = 2 radians
Fill in each unit circle with the degree
and radian measure for each line.
Final Exam
6th 83.8 87
7th 83.7 85.5
8th 85 89.5
Average Median
0
2
4
6
8
10
12
14
16
18
Too Slow Just Right Too Fast
Pace of Class
0
5
10
15
20
25
Too Easy Just Right Too Hard
Level of Difficulty
Section 7-1 Measurement of Angles
Objective: To find the measure of
an angle in either degrees or
radians.
Chapter 7
Trigonometric Functions
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.
• The terminal ray can be in any of the quadrants.
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–3
Section 4.1, Figure 4.2, Standard
Position of an Angle, pg. 248
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° –210°
Positive Angle: rotates counter-clockwise (CCW)
Negative Angle: rotates clockwise (CW)
When sketching
angles, always
use an arrow to
show direction.
Units of Angle Measurement
Degree
• 1/360th of a circle.
• This is the measure on a protractor and most
people are familiar with.
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
When the arc of circle has the same length asthe radius of the circle, angle measures 1 radian.
Arc Length = Radius
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?
Conversion Formulas: 360 2 radians
180 radians
o
o
To convert degrees to radians, multiply by 𝝅
𝟏𝟖𝟎
To convert radians to degrees, multiply by 𝟏𝟖𝟎
𝝅
Convert 196˚ to radians.
Convert 1.35 radians to degrees.
196˚∗𝜋
180˚=196𝜋
180=49𝜋
45radians
1.35 ∗180˚
𝜋= 77.35˚
Section 7-2
Sectors of CirclesObjective: To find the arc
length and area of a sector of a
circle and to solve problems
involving apparent size.
Sector of a CircleA sector of a circle is the region bounded
by a central angle and the intercepted arc.
Sector
A
B
s = arc length 𝐴𝐵
K= area of the sector
𝜃 = central angle
r = radius
Degrees
𝑠 =𝜃𝑟𝜋
180
𝐾 =𝜃𝑟2𝜋
360
𝑠 = 𝑟
𝐾 =1
2𝑟2
𝐾 =1
2𝑟𝑠
Radians
s = arc length
𝜃 = central angler = radius
K= area of the sector
Find the arc length and area of each sector.
Arc Length:
𝑠 = 𝜃𝑟
𝑠 =2𝜋
3∗ 6 = 4𝜋 in
Area: 𝐾 =1
2𝑟2𝜃
𝐾 =1
2∗ 62 ∗
2𝜋
3= 12𝜋 𝑖𝑛2
Arc Length:
𝒔 =𝜽𝒓𝝅𝟏𝟖𝟎
𝑠 =45∗4∗𝜋
180= 𝜋 cm
Area: 𝑲 =𝜽𝒓𝟐𝝅
𝟑𝟔𝟎
𝐾 =45 ∗ 42 ∗ 𝜋
360= 2𝜋𝑐𝑚2
6 25
s r
2A sector of a circle has arc length 6 cm and area = 75 cm .
Find its radius and the measure of its central .
1
2
25
175 6
2
K r
r
r
s
6s
75A
?r
?
60.24 Radians
25
1800.2 144
o
Apparent SizeHow big an object looks depends not only on its size
but also on the angle that it subtends at our eyes. The
measure of this angle is called the object’s apparent
size.
s
r
𝑠 = 𝑟 𝑠 =𝜃𝑟𝜋
180
Jupiter has an apparent size of 0.01° when it is
8 x 108 km from Earth. Find the approximate
diameter of Jupiter.
𝑠 =8 × 108 .01 𝜋
180𝑠 =𝜃𝑟𝜋
180= 139,626 km
Homework
Page 261 #1-11 odds
Page 264 #1-17 odds
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