Download Holt McDougal Algebra 2

The
TheUnit
UnitCircle
Circle
• How do we convert angle measures
between degrees and radians?
• How do we find the values of
trigonometric functions on the unit
circle?
HoltMcDougal
Algebra 2Algebra 2
Holt
The Unit Circle
You can use reference angles and Quadrant I of the
unit circle to determine the values of trigonometric
functions.
Trigonometric Functions and Reference Angles
Holt McDougal Algebra 2
The Unit Circle
Students
All
Take
Calculus
The diagram shows how
the signs of the
trigonometric functions
depend on the quadrant
containing the terminal
side of θ in standard
position.
Holt McDougal Algebra 2
The Unit Circle

0
radian 0
30 45 60 90
 6  4  3  2
0
2
0
11
2
2
2
2
3
2
1
cos 1
3
2
3
3
2
2
1
2
0
sin
tan 0
Holt McDougal Algebra 2
1
3
4
2
und.
The Unit Circle
Using Reference Angles to Evaluate Trigonometric Functions
Use a reference angle to find the exact value of the sine,
cosine, and tangent of each angle. Students
1. 330
All
o
Calculus
Take
360  330
o
30
Step 1 Find the reference angle.
Step 2 Find the sin, cos, and tan of the
reference angle.
Step 3 Adjust the signs, if needed.
1
sin 330  
2
o
3
cos 330 
2
3
tan 330  
3
o
Holt McDougal Algebra 2
o
The Unit Circle
Using Reference Angles to Evaluate Trigonometric Functions
Use a reference angle to find the exact value of the sine,
cosine, and tangent of each angle. Students
All
2. 270
270°
o
Take
Step 1 Find the reference angle.
Calculus
360  270
90 o
Step 2 Find the sin, cos, and tan of the
reference angle.
Step 3 Adjust the signs, if needed.
sin 90° = 1
o
cos 90° = 0
tan 90° = und.
Holt McDougal Algebra 2
sin 270  1
cos 270  0
o
tan 270o  und.
The Unit Circle
Using Reference Angles to Evaluate Trigonometric Functions
Use a reference angle to find the exact value of the sine,
cosine, and tangent of each angle. Students
All
11
3.
6
Step 1 Find the reference angle.
Take

Calculus
11
2 
6
6
Step 2 Find the sin, cos, and tan of the
reference angle.
Step 3 Adjust the signs, if needed.
 1
sin 
11
3
11
1
6 2
cos

sin


3
6
2
6
2
cos 
6
2
11
3
tan


3
6
3
tan 
6
3
Holt McDougal Algebra 2
The Unit Circle
Using Reference Angles to Evaluate Trigonometric Functions
Use a reference angle to find the exact value of the sine,
cosine, and tangent of each angle. Students 4
All
4
4. 
3
Step 1 Find the reference angle.
3
Take

Calculus
4

3
3
Step 2 Find the sin, cos, and tan of the
reference angle.
Step 3 Adjust the signs, if needed.

3
sin 
4 
1
4 
3


3
2
cos 
sin  


 1
2
 3 
 3  2
cos 
3 2
tan

3
 3
Holt McDougal Algebra 2
 4 
tan  
 3
 3 
The Unit Circle
If you know the measure of a central angle of a
circle, you can determine the length s of the arc
intercepted by the angle.
s  r
Holt McDougal Algebra 2
The Unit Circle
Automobile Application
5. A tire of a car makes 653 complete rotations in 1 min. The
diameter of the tire is 0.65 m. To the nearest meter, how far
does the car travel in 1 s?
Step 1 Find the radius of the tire.
The radius is
diameter.
of the
Step 2 Find the angle θ in radian through which the
tire rotates in 1 second.
1 rotation = 2.
653
653 2 1 min
1 min

1


60 sec30 30 sec
 653 
s  r  .325 m 
Change to seconds
  22.2 m/sec
 30 sec 
Holt McDougal Algebra 2
The Unit Circle
Automobile Application
6. An minute hand on Big Ben’s Clock Tower in London is 14
ft long. To the nearest tenth of a foot, how far does the tip of
the minute hand travel in 1 minute?
Step 1 Find the radius of the clock. r =14
Step 2 Find the angle θ in radian through which the
hour hand rotates in 1 minute.
1 hour = 2.

2
1 hour


1 hour 60 min 30 30 min
  
s  r  14 ft 

 30 min 
Holt McDougal Algebra 2
 1.5 ft/min
Change to minutes
The Unit Circle
Lesson 10.3 Practice B
Holt McDougal Algebra 2