PreCalc Polar coordinates and equations 6-4

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6/4/13 Obj: SWBAT plot polar coordinates
Bell Ringer: Plot the point (4, 4π/3)
HW:
Polar Coordinates WS
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(r, )
You are familiar with
plotting with a rectangular
coordinate system.
We are going to look at a
new coordinate system
called the polar
coordinate system.
The center of the graph is
called the pole.
Angles are measured from
the positive x axis.
Points are
represented by a
radius and an angle
radius
(r, )
To plot the point
  
 5, 
4 

First find the angle
Then move out along
the terminal side 5
A negative angle would be measured clockwise like usual.
3 

 3, 

4 

To plot a point with
a negative radius,
find the terminal
side of the angle
but then measure
from the pole in
the negative
direction of the
terminal side.
2 

  4,

3 

Let's plot the following points:
  
 7, 
2 

 

  7 ,

2 

3 
 5  

 7,
  7 ,
2 
2  

Notice unlike in the
rectangular
coordinate system,
there are many
ways to list the
same point.
Let's take a point in the rectangular coordinate system
and convert it to the polar coordinate system.
(3, 4)
r
4

3
Based on the trig you
know can you see
how to find r and ?
3 4 r
2
2
2
r=5
tan  
3
We'll find  in radians
polar coordinates are:
(5, 0.93)
  tan
4
1
4
   0 . 93
3
Let's generalize this to find formulas for converting from
rectangular to polar coordinates.
(x, y)
r

y
x  y r
2
r 
2
2
x  y
2
2
x
tan  
y
x
You need to consider the quadrant in
which P lies in order to find the value of .
  tan
1
 y
 
 x
Now let's go the other way, from polar to rectangular
coordinates.
Based on the trig you
know can you see
  
how to find x and y?
 4, 

4

4 
y

cos

4
4
x
x
4
 2
2 2
x  4
 2 


sin
rectangular coordinates are:
 2
2


,
 2

2



4

y
4
 2
2 2
y  4
 2 


Let's generalize the conversion from polar to rectangular
coordinates.
cos  
r ,  
r
 y
x
x
r
x  r cos 
sin  
y
r
y  r sin 
Polar coordinates can also be given with the angle in
degrees.
120
(8, 210°)
90
60
45
135
30
150
(6, -120°)
180
0
330
210
315
225
(-5, 300°)
300
240
270
(-3, 540°)
To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example
  
 4, 
 3 
12
13
Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions. Think about the
Pythagorean Theorem.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.
(5, 4)
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Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by
r 
x  y
2
2
 ref  tan
1
y
x
You need to consider the quadrant in
which P lies in order to find the value of .
P
15
Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.
a)
 2,  2 
16
Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.
b)
  1,  3 
17
The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.
5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)
7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )
Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
18
Convert the rectangular coordinate system equation to a
polar coordinate system equation.
From
conversion
s, how
2
2
x  y  9
2
2
r 
x  y
r  3
2
2
was r related to x and y ?
Here each r
unit is 1/2 and
we went out 3
and did all
angles.
Before we do the conversion
let's look at the graph.
r must be  3 but there is no
restriction on  so consider
all values.
Convert the rectangular coordinate system equation to a
polar coordinate system equation.
2
x  4y
What are the polar conversions
we found for x and y?
substitute in for
x and y
x  r cos 
 r cos  
y  r sin 
2
 4  r sin  
r cos   4 r sin 
2
2
We wouldn't recognize what this equation looked like
in polar coordinates but looking at the rectangular
equation we'd know it was a parabola.
Write Polar Equation in Rectangular Form
Given r = 2 sin θ
x  r co s 
– Write as rectangular
equation
y  r sin 
y
Use definitions
– And identities
 tan
1

x
r  x  y
2
2
2
– Graph the given equation for clues
21
Write Polar Equation in Rectangular Form
Given r = 2 sin θ
– We know
– Thus
– And
r
 sin  
2
y
r
r  2y
2
x  y  2y
2
2
22
Write Rectangular Equation in Polar Form
Consider 2x – 3y = 6
x  r co s 
y  r sin 
– As before, use
definitions
y
 tan
1

x
2  r cos   3  r sin   6
r  x  y
2
2
2
r  2 cos   3 sin    6
r 
6
 2 cos 
 3 sin 

23
Polar Coordinates
How do we graph or use
a graph to describe loops
and curves?
Background
The use of polar coordinates allows for
the analysis of families of curves difficult
to handle through rectangular coordinates (x,y). If a curve is a
rectangular coordinate graph of a function, it cannot have any
loops since, for a given value there can be at most one
corresponding value. However, using polar coordinates,
curves with loops can appear as graphs of functions.
http://www.xpmath.com/careers/topicsresult.php
?subjectID=4&topicID=18
24
25
Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.
•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.
26
Example
 
Plot the point P with polar coordinates  2,
 4

.

27
Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .
For r  0 , r = |r| and ϴ = ϴ + 180
Ex. (-7, 70⁰) = (7, 250)
28
Example
Plot the point with polar coordinates
  
 4, 
 3 
29
Plotting Points Using Polar Coordinates
a)
 5 
 3,

3 

b)
 

 2,  
4

30
Plotting Points Using Polar Coordinates
c)
 3, 0 
d)
 

  5, 
2

31
A)
B)
C)
D)
32
•
Using polar coordinates, the same point can
be described by many different
representations. Which of the following
do(does) not describe the point (8,60°) ?
(8,−60°) (8,-300°) (8,420°) (-8,-120°)
•
Using polar coordinates, write 3 more
representations of the point (5,150°)
33
Graphing in polar coordinates
•
Hit the MODE key.
•
Arrow down to where it says Func
(short for "function" which is a bit misleading since they are all functions).
•
Now, use the right arrow to choose Pol.
•
Hit ENTER.
(*It's easy to forget this step, but it's crucial: until you hit ENTER you have not actually
selected Pol, even though it looks like you have!)
The calculator is now in polar coordinates mode. To see what that means, try this.
 Hit the Y= key. Note that, instead of Y1=, Y2=, and so on, you now have r1= and so on.
 In the r1= slot, type 5-5sin(
 Now hit the familiar X,T,q,n key, and you get an unfamiliar result. In polar coordinates mode, this
key gives you a ϴ instead of an X.
 Finally, close off the parentheses and hit GRAPH.
If you did everything right, you just asked the calculator to graph the polar equation r=5-5sin(ϴ).
The result looks a bit like a valentine.
The WINDOW options are a little different in this mode too. You can still specify X and Y ranges,
which define the viewing screen. But you can also specify the ϴvalues that the calculator begins
and ends with; for instance, you may limit the graph to 0< ϴ <π/2. This would not change the
viewing window, but it would only draw part of the graph.
34
End here 6/3
35
End of Section
36
Definitions of Trig Functions of Any Angle
(Sect 8.1)
Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y
sin  
cos  
tan  
y
csc  
r
r
y
x
r
(x, y)
r
x
r
y
x
x
sec  
cot  

x
y
37
The Signs of the Trig Functions
Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.
38
The Signs of the Trig Functions
39
Where each trig function is POSITIVE:
“All Students Take Calculus”
Translation:
S
T
A
C
A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4
*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.
**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
40
wherever tangent is positive.
Example
Determine if the following functions are positive or negative:
sin 210°
cos 320°
cot (-135°)
csc 500°
tan 315°
41
Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614
2) tan    2.553
3) cos    0.866
42
Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0
2) sec   0, cot  0
43
Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).
44
Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles (Quads II, III, IV)
can be found using the values of the corresponding reference angles.
Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
45
Example
Find the reference angle for   2 2 5 
Solution
y
By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.
 ref  225   180 

 ref
x
 ref  45 
46
So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.
For example,
sin 225    (sin 45  )  
1
2
In Quad 3, sin is negative
45° is the ref angle
47
Example
Give the exact value of the trig function (without using a calculator).
cos 150
48
Examples (Text p 239 #6 & 8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.
6) tan 91 , sec 345 
8) cos 190 , cot 290 
49
Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the answer with the correct sign.
Remember:

Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree

To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .
50
Example
Evaluate co t 3 2 4 .0  . Round appropriately.

Set Mode to Degree

Enter: 
OR

A N S : cot 324.0    1.38
51
HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
y
Example:
Find the value of  to the nearest 0.01°

-12
x
r
-5
(-12, -5)
52
Examples
F in d  fo r 0     3 6 0 
1) sin    0.418
2) tan   1.058
53
Examples
F in d  fo r 0     3 6 0 
3) cos    0.85
4) cot    0.012, sin   0
54
BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.
2 sin   1  0
55
SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.
4(sin  )  9   6
2
56
Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...) ,
we will use the circle.
(0, 1)
90
Unit Circle:

Center (0, 0)

radius = 1

x2 + y2 = 1
(-1, 0)
(1, 0)
180
0
270
(0, -1)
57
Now using the definitions of the trig functions with r = 1,
we have:
sin  
cos  
y
 y
csc  
r

1
r
1
y
y
x
x
r
1
r
tan  

y
y
x

x
sec  
1
x
cot  

x
x
y
58
Example
Find the value of the six trig functions for    9 0 
(0, 1)
sin   90   
270
(-1, 0)
(1, 0)
180

90
(0, -1)
0
cos   90   
tan   90   
y

y
r
1
x
x

r
y


1

x
csc   90   
r
y
y
sec   90   
r
1
cot   90   
x


x
1


x

y
59
Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   
y

x
csc  0   
1
sec  0   
1
co t  0   
x

y

x

y
60
Example
Find the value of the six trig functions for 
 540 
sin  540    y 
cos  540    x 
tan  540   
y

x
csc  540   
1
sec  540   
1
cot  540   
x

y

x

y
61
Radian Measure (Sect 8.3)
A second way to measure angles is in radians.
Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,
 
s
r
62
Radian Measure
2 radians corresponds to 360 
2  6.28
 radians corresponds to 180 
  3.14


2
radians corresponds to 90 
 1.57
2
63
Radian Measure
64
Conversions Between Degrees and Radians
1.
2.
To convert degrees to radians, multiply degrees by
To convert radians to degrees, multiply radians by

180
180

Example
Convert from degrees to radians: 210º
210 
65
Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3
 3
4

4
b) Convert from radians to degrees: 3.8
3 .8 
66
Conversions Between Degrees and Radians
c) Convert from degrees to radians (exact):  6 7 5 
675 
d) Convert from radians to degrees:
13
13
6

6
67
Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 
f) Convert from radians to degrees (to nearest tenth): 1 rad
1
68
Examples
Find  to 4 sig digits for 0    2 
sin   0.9540
69

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