10 1 polar coordinates
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10.1 Polar Coordinates. The Cartesian system of rectangular coordinates is not the only graphing system. This chapter explores the polar coordinate system. P. ( r , θ ). r. θ. O. (polar axis). fixed ray OA. A. We will graph in what is called the r θ -plane.

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10.1 Polar Coordinates

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10 1 polar coordinates

10.1 Polar Coordinates


10 1 polar coordinates

The Cartesian system of rectangular coordinates is not the only graphing system. This chapter explores the polarcoordinatesystem.

P

(r, θ)

r

θ

O

(polar axis)

fixed ray OA

A

We will graph in what is called the rθ-plane

fixed point (pole or origin)

A polar coordinate is the ordered pair (r, θ)

r = distance from pole to point

θ = angle from polar axis (deg or rad)

(pos or neg)

on

terminal side

on opposite of terminal side

(pos or neg)

counterclockwise

clockwise


10 1 polar coordinates

Ex 1) Graph each point on the rθ-plane. (Just sketch)

a)

b)

c)

O

P

Q

O

O

R


10 1 polar coordinates

Note: Since θ and θ + 2πn, n  will produce equal angles, a point can be represented in infinitely many polar coordinate pairs.

r can also be positive or negative, adding to the options

Note: If r > 0 and 0 ≤ θ < 2π, then (r, θ) represents exactly 1 point.

Ex 2) Plot

Which of these does NOT represent the same point?

(Identify and fix it)

1

A)B) C)


10 1 polar coordinates

An equation with polar coordinates is a polarequation. We will graph with constants today, r = c and θ = k, and explore more complicated ones tomorrow.

Ex 3) Graph each polar equation.

a) r = 3

(length always 3

angle is anything)

1 2 3

b)

(angle always

r can be anything

positive or negative)

OR


10 1 polar coordinates

Your turn. Graph on whiteboard.

c)d) r = –4

*same as r = 4

1 2 3 4


10 1 polar coordinates

If we superimposed the rectangular coordinate system on the rθ-plane we can discover their relationships.

In cartesian:

(x, y)

(r, θ)

r

y

= r sinθ

θ

x

= r cosθ

and

also

You will use these relationships to change equations from one system to another system.


10 1 polar coordinates

Ex 4) Find the rectangular coordinates. Round to nearest hundredth.

(if necessary)

a)

b)

(not famous – use calculator)

**RAD mode


10 1 polar coordinates

To convert from rectangular to polar:

(if x > 0)

(if x < 0)

Ex 5) Find polar coordinates of


10 1 polar coordinates

Ex 6) Convert:

a) x = 3 to a polar equation

b)to a rectangular equation

x2 + y2

y


Homework

Homework

#1001 Pg 482 #1–53 odd, 34, 40, 54


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