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)