9-3
Study Guide Polar and Rectangular Coordinates
Converting
Polar
Coordinates to
Rectangular
Coordinates
Example 1
The rectangular coordinates (x, y) of a point named by the polar
coordinates (r, ) can be found by using the following formulas.
x r cos y r sin Find the rectangular coordinates of each point.
3
a. P3, 4 b. Q(20, 60°)
3
3
, r 3 and .
For P3, 4 4
Use the conversion formulas
x r cos and y r sin .
x r cos 3
3 cos 4
2
3
2 x r cos 20cos (60°)
20(0.5)
10
y r sin 3
3 sin 4
3
2
or 2
The rectangular coordinates
3
2, 3
2
, or
of P are 2
2 (2.12, 2.12) to the nearest
hundredth.
Example 2
y r sin 20 sin(60°)
3
20
2 103
The rectangular coordinates of Q are
(10, 103 ), or approximately
(10, 17.32)
2
3
2 3
2
or 2
For Q(20, 60°), r 20 and
60°.
Converting
Rectangular
Coordinates
to Polar
Coordinates
The polar coordinates (r, ) of a point named by the rectangular
coordinates (x, y) can be found by the following formulas.
r
y
x
Arctan , when x 0
For R(5, 9), x 5 and y 9.
y
Arctan x
(9
)2
52
1
06
or about 10.30
x0
9
Arctan 5
1.06
To obtain an angle between 0 and 2 you can add
2 to the -value. This results in 5.22.
The polar coordinates of R are approximately
(10.30, 5.22).
Example 3
Write the polar equation r 5 cos in
rectangular form.
r 5 cos r2 5r cos x2 y2 5x
y
x
Arctan , when x 0
Find the polar coordinates of R(5, 9).
2 r x
y2
2 苶
y2
兹x苶
Multiply each side by r.
r 2 x 2 y 2 and r cos x
9-3
Precalculus B 2017
Polar and Rectangular Coordinates
Find the rectangular coordinates of each point with the
given polar coordinates.
1. (6, 120)
2. (4, 45)
3. 4, 6
13
4. 0, 3 Find the polar coordinates of each point with the given
rectangular coordinates. Use 0 2 and r 0.
5. (2, 2)
6. (2, 3)
)
7. (3, 3
8. (5, 8)
Write each polar equation in rectangular form.
9. r 4
10. r cos 5
Write each rectangular equation in polar form.
12. y 3
11. x2 y2 9
13. Surveying A surveyor records the polar coordinates of the location
of a landmark as (40, 62°). What are the rectangular coordinates?