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AP Calculus BC Unit 1 Day 2
Arrival Activity
• With a partner, discuss any issues you had with last night’s homework
• AND answer the following using the same graph that was given for HW #1-8
▫ What is the point that is located at –π/6 ?
• QUIZ today!!!!
HW Questions
Next . . . . . • Continuation of yesterday’s notes/topic
Referring to yesterday’s notes, what do the
graphs of the following look like.
3
1r
Can you think of another equation that would result in the same graph?
Can you think of two other equations that would result in the same graph?
What would the graph of these pairs of
equations look like?
. 1 2 0
2A r and
. 3 24
B r and
How about this pair ?
. 04
C r and
AP Calculus BC NEW MATERIAL
Polar vs Cartesian
Unit 1 Day 2
2
2 2 2 2 2 2
2 2
4 4 2 2 3 2 2
cos 2 2
cos sin 4 4
cos sin 1 1
1 2 cos 3 4 1 0
1 cos 2 2 2 0
Polar Cartesian
r x
r xy
r r x y
r r y x x
r x y x y x xy y
It is good to have a Cartesian coordinate system AND a Polar
coordinate system because some functions are easier in
Cartesian and some are easier in Polar:
Recall from yesterday . . . .
r
,r
Same point but different
coordinate system . . .
,x y
y
x
r
,r
Converting from polar to Cartesian:
cosx r
siny r
Example:
Convert to Cartesian coordinates :
4,6
,x y
Now, YOU try!!
Convert into Cartesian:
6,5)3
4
5,3)2
3
7,4)1
2
5,
2
35)3
2
23,
2
23)2
32,2)1
:Answers
Converting from Cartesian to Polar
,x y
y
x
tany
x 222 yxr
Example:
Convert to Polar
Coordinates
Finding r:
Finding is more involved . . .
3,3
Finding . . . .
tany
x
222 yxr
Example:
Convert to Polar Coordinates
Now we have to pair up the r and . . .
3,3
Pairing up r and . . .
The original Cartesian coordinates place the point in the 2nd quadrant.
So . . . .
12r 11 5
or 6 6
The Cartesian point in Polar
form is:
3,3
6
5,32
112 3,
6
There are multiple conversions to polar!
Being in the correct location when finished with the conversion is what is important.
OR
Up Next . . .
Converting points that would not have a landing on a known locations on the unit circle
Convert to Polar Coordinates
1 4tan 0.927
3
The other possibility would be to add to the angle
and use the positive value of r.
2 2
3 4 5r
3, 4
This is in the first quadrant.
To end up in the 3rd quadrant, the location of the given
point, we would need 5,0.927
5,0.927
Practice:
Convert into Polar Coordinates
2,2)3
1,1)2
3,1)1
:
( )
1) 2,3
2) 2,4
73) 2,
4
Answers
at least one of them
Recap
r
y
x
222 yxr
tany
x
,
,
r
x y
sin sin y
y rr
cos cos x
x rr
Cartesian to Polar
Polar to Cartesian
Converting EQUATIONS from Polar to Cartesian
1. 3secr
Confirm your answer by graphing the original polar equation to see that it is a vertical line at x=-3
52.
sin 2cosr
Confirm your answer by graphing the original polar equation to see that it is a line equivalent to y=5+2x
QUIZ--Unit Circle BC Style
• AFTER quiz start on HW
▫ Textbook pg. 738 (1, 3, 5, 11, 13, 15, 27, 31, 33, 37, 41, and 45)