Transcript Polar Coordinates Polar and Rectangular

Polar Coordinates
Polar and Rectangular
Coordinates
Packet 3
Polar vs. Rectangular
Coordinates

For some real-world phenomena, it is
useful to be able to convert between polar
coordinates and rectangular coordinates.
Polar vs. Rectangular
Coordinates

Suppose a rectangular
coordinate system is
superimposed on a
polar coordinate system
so that the origins
coincide and the x-axis
aligns with the polar
axis, as shown at the
right. Let P be any
point in the plane.


Polar Coordinates: P(r, θ)
Rectangular Coordinates: P(x, y)
Polar to Rectangular
Coordinates
Trigonometric functions can be used to
convert polar coordinates to rectangular
coordinates.
 The rectangular coordinates (x, y) of a
point named by the polar coordinates (r, θ)
can be found by using the following
formulas:
x = r cos θ
y = r sin θ

Rectangular to Polar
Coordinates

If a point is named by the rectangular
coordinates (x, y), you can find the
corresponding polar coordinates by using
the Pythagorean Theorem and the
Arctangent function (Arctangent is also
known as the inverse tangent function).
Rectangular to Polar
Coordinates

Since the Arctangent function only
determines angles in quadrants 1 and 4
(because tangent has an inverse in
quadrants 1 and 4) you must add π
radians to the value of θ for points with
coordinates (x, y) that are in quadrants 2
or 3.
Rectangular to Polar
Coordinates

When x > 0, θ = Tan
1
y
x

When x < 0, θ = Tan
1
y

x
Rectangular to Polar Coordinates



When x is zero,    . Why?
2
The polar coordinates (r, θ) of a point named by
the rectangular coordinates (x, y) can be found
by the following formulas:
r = x2  y2
1 y
When x > 0   Tan
x
1 y

When x < 0   Tan
x
Converting Equations
The conversion equations can also be
used to convert equations from one
coordinate system to the other.
 Look at examples 4 and 5

Examples
Find the rectangular coordinates of each point.
a.
5 

D  2,

6


b. C(3, 270°)
Example
Find the polar coordinates of E(2, -4)
Write the polar equation r = -3 in rectangular
form.
Example
Write the rectangular equation x² + (y – 1)² = 1
in polar form.