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