P.o.D. – Write each parametric
equation in rectangular form.
1.) x=3-2t, y=2+3t
1
2.) 𝑥 = 𝑡; 𝑦 = 𝑡 2
4
3.) 𝑥 = 𝑡 + 2, 𝑦 = 𝑡 2
4.) 𝑥 = √𝑡, 𝑦 = 1 − 𝑡
5.) 𝑥 = 𝑡 − 1, 𝑦 =
𝑡
𝑡−1
6.) 𝑥 = 3 cos 𝜃, 𝑦 = 3 sin 𝜃
1.) 𝑦 =
−3
2
𝑥+
13
2
2.) 𝑦 = 16𝑥 2
3.) 𝑦 = 𝑥 2 − 4𝑥 + 4
4.) 𝑦 = 1 − 𝑥 2
5.) 𝑦 =
6.)
𝑥2
9
+
𝑥+1
𝑥
𝑦2
9
=1
10.7 – Polar Coordinates
Learning Target(s): I can plot
points on the polar coordinate
system; convert points from
rectangular to polar form and
vice versa; convert equations
from rectangular to polar form
and vice versa.
The Polar Coordinate System:
- We can graph points using
their radius and central angle
(𝑟, 𝜃).
Plot the following points:
a.)
b.)
c.)
d.)
(2,60°)
(3, −120°)
(−2,45°)
(−3, −60°)
(draw the graphs on the
whiteboard)
- We can also write coterminal
angles in polar form.
For example, (−3, −60°) =
(−3, −60° + 360°) = (−3,300°) =
(3,120°) by using the positive
radius and the supplement.
EX: Give 3 points that are
coterminal with (2,45°).
(plot the coordinate on the
board)
(2, −315°) = (2,405°) = (−2, −135°)
= (−2,225°) = 𝑒𝑡𝑐.
Converting Rectangular to Polar
Coordinates:
Given (x,y)  (𝑟, 𝜃) where 𝑟 =
±√𝑥 2
+
𝑦2
and 𝜃 =
−1 𝑦
tan ( )
𝑥
EX: Convert the following
points to polar coordinates:
a.) (2,2)
b.) (1,3)
a.) 𝑟 = √22 + 22 = √8 = 2√2
𝜃 = tan
−1
2
( ) = tan−1 1 = 45°
2
(2√2, 45°)
b.) 𝑟 = √1 + 9 = √10
𝜃 = tan−1 3 = 71.57°
(√10, 71.57°)
Converting Polar Coordinates to
Rectangular:
Given (𝑟, 𝜃)  (x,y) where 𝑥 =
𝑟 cos 𝜃 and 𝑦 = 𝑟 sin 𝜃
EX: Convert the following
Rectangular coordinates to
Polar form:
a.) (5,30°)
b.) (−2, −41°)
a.) 𝑥 = 5 cos 30° = 5 ∙
√3
2
=
5√3
2
1 5
𝑦 = 5 sin 30° = 5 ∙ =
2 2
5√3 5
(
, )
2 2
b.) 𝑥 = −2 cos(−41°) = −1.51
𝑦 = −2 sin(−41°) = 1.31
(−1.51,1.31)
*We can do these conversions on
the calculator.
EX: Plot the given point in the
polar coordinate system:
a.) (3,
−𝜋
3
𝜋
)
b.) (2, )
c.) (2,
6
−𝜋
6
)
(show the graphs on the
whiteboard)
EX: Find 3 additional polar
representations of (2,
3𝜋
4
).
5𝜋
𝜋
7𝜋
(2, − ) , (−2, − ) , (−2, )
4
4
4
EX: Convert each point to
rectangular coordinates:
a.) (4,
3𝜋
6
𝜋
)
b.) (2, )
6
a.) (0,4)
b.) (√3, 1)
EX: Convert each point to polar
coordinates, in radians:
a.) (-2,2)
b.) (-1,0)
a.) (2√2,
b.) (1, 𝜋)
3𝜋
4
)
EX: Describe the graph of each
polar equation and find the
corresponding rectangular
equation.
a.) r=1
𝜋
b.) 𝜃 =
4
c.) 𝑟 = csc 𝜃
We will want to examine each
polar graph first.
a.)
This is obviously
a circle with a radius of 1
(the Unit Circle). We
simply need to write this
equation in rectangular
form. 𝑥 2 + 𝑦 2 = 1.
b.) This is simply a line at an
𝜋
angle of radians or 45
4
degrees. (draw a graph on
the whiteboard). What line
has an inclination of 45
degrees? Y=x.
c.)
This is a
horizontal line at 1, so the
equation is y=1.
Coordinate Conversion
Polar-to-Rectangular:
𝑥 = 𝑟 cos 𝜃
𝑦 = 𝑟 sin 𝜃
Rectangular-to-Polar:
𝑦
tan 𝜃 =
𝑥
𝑟2 = 𝑥2 + 𝑦2
EX: Convert the rectangular
equation 𝑥 2 + 𝑦 2 = 9𝑎2 into polar
form.
𝑟 2 = 9𝑎2
𝑟 = 3𝑎
Upon completion of this lesson,
you should be able to:
1. Graph polar coordinates.
2. Convert from rectangular
coordinates to polar and vice
versa.
For more information, visit
http://www.mathsisfun.com/polar-cartesiancoordinates.html
HW Pg.783 3-63 3rds, 77-80