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LESSON 6.1 – ANGLE MEASURE
Trigonometry
Measurement of triangles
Angles
An angle is in standard position if…
1. its initial side is along the positive x-axis
2. its vertex is at the origin, and
Note: Labeled with
Greek letters:
 ,  ,  ,...
Coterminal Angles
Angles with the same initial and terminal sides.
 and   360n
 and   2 n
Review
1. Central angle
2. Acute angle
3. Right angle
4. Obtuse angle
5. Arc measure
6. Arc length
7. Complementary angles
8. Supplementary angles
Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two
negative angles that are co-terminal with the given angle. Sketch the angles.
1.
210
2.
45
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 3
3.
540
Radian Measure
One radian
  is the measure of a central
Note: In a full
revolution, the arc
length s is equal to
C  2 r  s .
Also, there are just
over six radius lengths
in a full circle.
Therefore, the central
angle  that intercepts an arc s equal in
length to the radius of a circle.
C = 2r  6.28r
The radian measure of an angle of one full
revolution is 2 . Since one full circle has
360 ,
360  2
180  
rad
Degrees to Radians
Radians to Degrees
Multiply by
Multiply by
rad
90 

2

angle is 


s
r
where
is measured in
radians.
60 
rad

3
rad
Example:
180
180
Example:

Practice Problems: Convert the following angles from degrees to radians and from radians to degrees without
using a calculator.
4.
150 
5.
7

6
6.
240 
7.

11

30
Practice Problems: Convert the following angles from degrees to radians and from radians to degrees using a
calculator and round to 3 decimal places.
8.
87.4 
9.
2 
10.
0.54 
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 4
11.
0.57 
Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two
negative angles that are co-terminal with the given angle. Sketch the angles.
12.
13
6
13.
3
4
14.
In a circle of radius r, the length s of an arc that
subtends a central angle of  radians is: s  r
Length of a
Circular Arc
2
3
Example:
Note:  must be in radians.
Review Problem 15: Find the following arc lengths using geometry then use s  r to validate your
answers.
CB  24in ,
mA0 B  60 , find the
Given
1. Circumference =
following arc lengths using 2
methods
2. Length of
AB
3. Length of
=
CA
4. Length of
CDB
=
5. Length of
ADB
=
6. Length of
ADC
=
=
A
60
O
C
B
24 in
D
m
AB 

mCA



mCDB
m
ADB 

m
ADC 


Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 5
Practice Problems: Find the unknown value.
16.
A central angle  in a circle of radius 24 cm is
subtended by an arc of length 6 cm. Find the measure
of  in radians.
17.
18.
A bicycle’s wheels are 14 inches in diameter.
How far (in miles) will the bike travel if its wheels
revolve 500 times without slipping?
19.
How many revolutions will a Ferris wheel of
diameter 60 feet make as the Ferris wheel travels a
distance of a ½ mile?
20.
An ant is sitting 5 cm from the center of a c.d..
If the c.d. turns 40  , how far has the ant moved in
meters?
21.
A bug is on a car’s windshield wiper and is 10
inches from the base of the windshield wiper. If the
bug moves 34 inches, at what angle did the windshield
wiper turn?
Find the radius of the circle if an arc of length
8 in on the circle subtends a central angle of

4
.

The angular velocity of a point on a rotating object is the
number of degrees (radians, revolutions, etc.) per unit
time through which the point turns.

Linear Speed
The linear velocity of a point on a rotating object is the
distance per unit time that the point travels along its
circular path.
v
Note:
The linear velocity depends on how far the object is from the axis of rotation, whereas
the angular velocity is the same no matter where the object lies on the rotating object.
Angular Speed
t
s
t
Relationship
If a point moves along a circle of radius r with angular speed  , then its linear speed v
between Linear and is given by: v  r
Angular Speed
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 6
Practice Problems: Solve the following problems.
22.
A woman is riding a bike whose wheels are 26
inches in diameter. If the wheels rotate at 125
revolutions per minute (rpm), find the speed at which
she is traveling, in miles per hour.
23.
The rear wheels of a tractor are 4 feet in
diameter, and turn at 20 rpm.
(a) How fast is the tractor going (feet per second)?
(b) The front wheels have a diameter of only 1.8 feet.
What is the linear velocity of a point on their tire
treads?
(c) What is the angular velocity of the front wheels in
rpm?
24.
The pedals on a bike turn the front sprocket at
8 radians per second. The sprocket has a diameter of
20 cm. The back sprocket, connected to the wheel, has
a diameter of 6 cm.
25.
Dan and Ella are riding on a Ferris wheel. Dan
observes that it takes 20 seconds to make
a complete revolution. Their seat is 25 feet from the
axle of the wheel.
(a) Find the linear velocity of the chain.
(a) What is their angular velocity?
(b) Find the angular velocity of the back sprocket.
(b) What is their linear velocity?
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 7
LESSON 6.2 – TRIGONOMETRY OF RIGHT TRIANGLES
The Trigonometric
Ratios
Let  has an acute angle of a right triangle. The
six trigonometric functions of the angle  are
defined below.
sin  
opp
hyp
csc 
hyp
opp
cos 
adj
hyp
sec 
hyp
adj
tan  
opp
adj
cot  
adj
opp
Review: Special
Right Triangles
45
30
x 2
x
SOH CAH TOA
2x
x 3
45
90
x
90
60
x
Practice Review Problems: Evaluate the following
1.
a4
b
c
2.
a
b6 2
c
3.
a
b
c  10
4.
a7 3
b
c
5.
a
b  hat
c
6.
a
b
c  iPod 7
45
c
b
45
90
a
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 8
7.
a4
b
c
8.
a
b6 2
c
9.
a
b
c  10
10.
a7 3
b
c
11.
a
b  hat
c
12.
a
b
c  iPod 7
30
b
c
90
60
a
Practice Problems: Evaluate the six trig functions at each real number without using a calculator.
1.
2.
sin  
csc 
17
4
cos 
sec 
tan  
cot  
sin  
csc 
cos 
sec  6
tan  
cot  
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 9