Download What are the coordinates?

Start Up
Day 22
1. Determine the exact values of x and y in each of
the following:
°
60
1
1 45
45
30
2. Convert each of the following angle measures:
135
°
210
°
5
4
11p
6
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
1
OBJECTIVE: SWBAT find the trigonometric
functions of the unit circle without a
calculator. SWBAT evaluate the
trigonometric functions of any angle.
EQ: How are the special values on the
unit circle determined and how can the
unit circle help us evaluate trig
functions quickly?
HOME LEARNING: p. 347 #21-42
All, 67 and 70 + MATHXL QUIZ
(4.1 & 4.2)
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2
Demana, Waits, Foley, Kennedy
4.3
Trigonometric Functions
of Any Angle
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3
Angles within the coordinate plane

Our “NEW”
definition

An ANGLE is
determined by
rotating a ray
about its
endpoint.

The INITIAL SIDE of
an angle is the
starting position of
the ray and the
TERMINAL SIDE is
the position after
rotation.
INITIAL SIDE
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4
STANDARD POSITION




An Angle in STANDARD
POSITION is an angle that
fits into the coordinate plane.
The ORIGIN is the VERTEX
The INITIAL SIDE coincides
with the positive x-axis
The TERMINAL SIDE holds
the position of the rotation.
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
INITIAL SIDE
5
Angles—positive and negatives?
0

By placing our Angles in
the Coordinate Plane
we allow the possibility
of both POSITIVE
(rotating “up” or
“counter-clockwise”)
and NEGATIVE
(rotating “down” or
“clockwise”) ANGLES
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0
6
ANGLES—
OH, THE POSSIBILITIES!

Could you ever have imagined an angle less than

Have you ever thought of an angle bigger than
?
360
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7
CAN YOU SEE THE DEGREE?
580
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8
Coterminal Angles
Two angles in an extended angle-measurement
system can have the same initial side and the
same terminal side, yet have different measures.
Such angles are called coterminal angles.
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9
Example: Finding Coterminal Angles
Find a positive angle and a negative angle that are
coterminal with 45º.
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10
Solution
Find a positive angle and a negative angle that are
coterminal with 45º.
Add 360º: 45º 360º  405º
Subtract 360º: 45º 360º  315º
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11
Example: Finding Coterminal Angles
Find a positive angle and a negative angle that are
coterminal with

6
.
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12
Solution
Find a positive angle and a negative angle that are
coterminal with

6
.

13
Add 2 :  2 
6
6

11
Subtract 2 :  2  
6
6
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13
What does “unit circle” really mean?
It’s a circle with a radius of 1
unit.
What is the equation of the “unit circle”?
x  y 1
2
2
Let’s make our own“unit circle”!
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14
Fold
your
paper
plate
i
half
twice!
Now,
 , 180
use a
-1,0
ruler
to
retrac
e
your
axes

0,1
2
90°
0, 0
1,0
2 , 360
270°
0, -1
3
2
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15
Here is the unit circle divided into 8 pieces. Fold your plate
one more time. Can you figure out how many degrees are
in each
0,1division?

9
0
°
135
°
45
°
18
 1,0 0°
225
°
0 1,0
°
27
0°
0,1
315
°
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16
Here is the unit circle divided into 12 pieces. It could be
folded, but might be easier to use a protractor to mark these
angles. Can you figure out how many degrees are in each
division?
0,1
12 9
0 60
0°
15
°
°
0°
30
°
18
 1,0 0°
0 1,0
°
21
0°
330
24
°
30
0° 27
0° 0°
0,1
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17
Let’s start with our special right
triangles….
1
60°
1
2
2
1
2
30
45
45
2
2
3
2
And apply them to the unit circle…
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18

Let’s begin with an easy family… 45° or
4
What are the
coordinates?

2

4
1
 , 180

2
2
,
2
2

2
2
45
2
0, 0
2 , 360
2
3
2
Now, reflect the triangle to the second quadrant…
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19
What are the
coordinates?

-
2
2
,
2
2


2

3
4
4
1
2
2
1
-
2
2
2
2
,
2
2

2
45
 , 180

2
0, 0
2 , 360
2
2
3
2
Now, reflect the triangle to the third quadrant…
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20

2
-
2
,
2
2


2
3
4
4
1
2
2
-

What are the
coordinates?
-
2
, -
2
2

1
2
2
2

2
2
,
2
2

2
45
 , 180
2

0, 0
2 , 360
2
2
5
4
3
2
Now, reflect the triangle to the fourth quadrant…
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21

2
-
2
,
2
2


2
3
4
4
1
2
2
-

-
2
, -
2
2

1
,
2
7
3
2
2
2

0, 0
2 , 360
What are the
coordinates?
2
2
5
4
2
2
2

2
2
45
 , 180
2

4

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2
2
, -
2
2

22

Complete the family…6
or 30°.

6

1
1
30
2
3
2
,
1
2

3
2
Now, reflect the triangle to the second
quadrant.
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23

3 1
,
2 2


6

5
6
1
2
-
3
2
1
1
30
2
3
2
,
1
2

3
2
Now, reflect the triangle to the third
quadrant.
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24

3 1
,
2 2

5
6
1
2
What are the
coordinates?

-
3
2
, -
1
2


6

-
7
3
2
1
1
30
2
3
2
,
1
2

3
2
6
Now, reflect the triangle to the fourth
quadrant.
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25

3 1
,
2 2

5
6
1
2
-

-
3
2
, -
1
2


6

7
3
2
1
1
30
2
3
2
,
1
2

3
2
11
6
6
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
3
2
, -
1
2

What are
coordinate
26

Let’s look at another “family”…60° or
3



2
3
2
,
3
2

3
1
2
 , 180
1
60
1
0, 0
2 , 360
2
3
2
Now, reflect the triangle to the second quadrant
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27
What are the
coordinates?

-
1
2
,
3
2




2
2
3
3
1
3
1
2
,
2
-
1
60
1
2
2

3
1
2
 , 180
3
0, 0
2 , 360
2
3
2
Now, reflect the triangle to the third
quadrant
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28



-
1
2
,
3
2

2
3
1
2
-

2
, -
2

2
,
3
2

4
3
3
1
2
 , 180
-
1
3
3
What are
the
coordinate
s? 1
3

2
1
60
1
2
0, 0
2 , 360
2
3
2
Now, reflect the triangle to the fourth
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29



-
1
2
3
,
2

2
3
1
2
-

-
2
, -
2

2
,
3
2

4
3
3
1
2
 , 180
3
1
3
3
1

2
1
0, 0
2 , 360
60
1
2
2
3
2
5
3
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.

What are the
coordinates?
1
2
, -
3
2

30
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31
Start Up Day 23
Too Close for Comfort: An F-15 aircraft
flying at an altitude of 8000 ft passes directly
over a group of vacationers hiking at 7400 ft. If
u is the angle of elevation from the hikers to
the F-15, find the distance d from the group to
the jet for the given angle.
(a) u = 45° (b) u = 90°
(c) u =
140°
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
32
OBJECTIVE: SWBAT find the trigonometric
functions of the unit circle without a
calculator. SWBAT evaluate the
trigonometric functions of any angle.
EQ: How are the special values on the unit
circle determined? How are the 6
trigonometric functions defined for a point
on the terminal side of angle in standard
position?
HOME LEARNING: p. 347 #1, 2, 3,
6, 7, 10, 11, 14, 16, 62, 66 & 68 +
Study your Unit Circle!
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33
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
34
The 3 Primary Trig Functions



SINE—The sine function is defined as the “ycoordinate” of a point at a specified rotation on
the UNIT CIRCLE
COSINE —The cosine function is defined as the
“x-coordinate” of a point at a specified rotation on
the UNIT CIRCLE
TANGENT—The Tangent function is defined as
the quotient of the”y-coord”/”x-coord” of a point
at a specified rotation on the UNIT CIRCLE
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35
Tangent--a bit tricky!
1
2
æ 5p ö y
tan ç ÷ = =
è 6 ø x - 3
2
1
2 2  1
3 2
3

2
1
3  3



3
3
3
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36
The More Connections You See, The
Easier it will Be!
  2
cos
 3
 2 
1
1

cos
 2
 3 
2

cos    cos
 
3
sin   
3
2
3
 
sin      3
2
sin      sin 
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37
Evaluate the six trigonometric
functions at each real number.
 1 3 
 ,

 2 2 


2

3
 2 
Sin 

 3 
 2 
Cos 

 3 
 2 
Tan 

3


= 
y
3
2
1
= 
2
x
y

x

3
1
3
2

 

2
1
2
2
 3
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38
The 3 RECIPROCAL FUNCTIONS



COSECANT—Sine’s Reciprocal Function
OR “y’s flip fraction”
SECANT –Cosine’s Reciprocal Function OR
“x’s flip fraction”
COTANGENT—Tangent’s Reciprocal
Function OR x/y
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39
Evaluate the six trigonometric functions at each real
number.
 1 3 
 ,

 2 2 


2

3
 2 
Sin 
 
 3 
 2 
Cos 

 3 
3
2
1

2
 2 
Tan 

3


 3
 2 
Csc 

 3 
2
2 3
3


3
3
3
 2 
Sec 

3


-2
 2
Cot 
 3
1




  3
3
 3

3
3
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40
Evaluate the six trigonometric functions at each real
number.


2
  
Sin 

 2 
  
Cos 

 2 
  
Tan 

2


(0, -1)
=
y
= -1
  
Csc 

 2 
=
x
=0
1
  
Sec 
 

 2 
0
y

x
1


0
0
  
Cot 
 
 2 
1
DNE
= -1
Does Not
Exist
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
DNE
=0
41
Denominator Driven DOMAIN
ISSUES



The Domain for Sine and Cosine is all real
numbers OR all angles.—no issues!
The Domain for Tangent and Secant is all real
numbers except any ODD multiples of π/2,
because the x-coordinate = ZERO there and
we CANNOT DIVIDE BY ZERO
The Domain for Cotangent and Cosecant is all
real numbers except integer multiples of π,
because the y-coordinate = ZERO there and
we CANNOT DIVIDE BY ZERO
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42
Let’s Apply it—Now you try it!
Evaluate the six trigonometric
7 functions of Θ

 7
Sin  4



7
Cos  
 4 
Ta
n
 7 


 4 
4
 2  2


 2 , 2 


7 

 4 
Csc
Se
c
Co
t
 7

 4
 2



2
 7 


 4 
-1
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
The More
Connections
you see…the
easier it will be!
43
Challenge: What is the sin 420°?
In fact sin 780° = sin
60° since that is just
another 360° beyond
420°.
Because the sine
values are equal for
coterminal angles
that are multiples of
360° added to an
angle, we say that
the sine is periodic
with a period of
360° or 2.
1
3
 ,

2

2


All the way around is 360° so we’ll need more than that. We
see that it will be the same as sin 60° since they are
44
coterminal angles. So sin 420° = sin 60°.
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
Trigonometric Functions of any Angle
Let  be any angle in standard position and let P(x, y)
be any point on the terminal side of the angle (except
the origin). Let r denote the distance from P(x, y)
to the origin, i.e., let r  x  y . Then
2
y
sin  
r
x
cos 
r
y
tan  
(x  0)
x
r
csc 
y
r
sec 
x
x
cot  
y
2
(y  0)
(x  0)
(y  0)
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
45
Example: Evaluating Trig Functions
Determined by a Point in QI
Let  be the acute angle in standard position whose
terminal side contains the point (3,5).
Find the six trigonometric functions of  .
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
46
Solution
Let  be the acute angle in standard position whose
terminal side contains the point (3,5).
Find the six trigonometric functions of  .
The distance from (3,5) to the origin is 34.
5
sin  
 0.857
34
34
csc 
 1.166
5
3
 0.514
34
34
 1.944
3
cos 
5
tan  
3
sec 
3
cot  
5
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
47
Example: Evaluating More Trig
Functions
Find sin 210º without a calculator.
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
48
Solution
Find sin 210º without a calculator.
An angle of 210º in standard position determines
a 30º–60º–90º reference triangle in the third quadrant.


The lengths of the sides stermines the point P  3,1 .
The hypotenuse is r  2.
y
1
sin 210º   
r
2
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49
Example: Using One Trig Ratio to
Find the Others
Find sin  and cot  by using the given information to
construct a reference triangle.
8
a. cos  
and csc  0
17
1
b. tan    and cos  0
2
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
50
Solution
8
a. cos  
and csc  0
17
1
Since cos  0 and csc 
0
sin 
the terminal side is in QIII.
Draw a reference triangle with
r  17, x  8.
and y  17 2  8 2  15
15
8
sin   
and cot  
17
15
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51
Solution Continued
1
b. tan   and cos  0
2
Since tan   0 and cos  0, the terminal side is in QIV.
Draw a reference triangle with
x  2, y  1.
and r  2 2  12  5
1
sin  
- -0.447 and
5
2
cot     2
1
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52