http://www.ck12.org Chapter 10. Polar and Parametric Equations
You can also express the relationship between x,yandrusing the Pythagorean Theorem.
x^2 +y^2 =r^2
Note that coordinates in polar form are not unique. This is because there are an infinite number of coterminal angles
that point towards any given(x,y)coordinate.
Once you can translate back and forth between points, use the same substitutions to change equations too. A polar
equation is written with the radius as a function of the angle. This means an equation in polar form should be written
in the formr=.
Example A
Convert the point (3, 4) to polar coordinates in three different ways.
Solution:tanθ=^43
θ=tan−^1( 4
3
)
≈ 53. 1 ◦
r^2 = 32 + 42
r= 5Three equivalent polar coordinates for the point (3, 4) are:
( 5 , 53. 1 ◦), ( 5 , 413. 1 ◦), (− 5 , 233. 1 ◦)
Notice how the third coordinate points in the opposite direction and has a seemingly negative radius. This means go
in the opposite direction of the angle.
Example B
Write the equation of the line in polar form:y=−x+ 1.
Solution: Make substitutions foryandx. Then, solve forr.
r·sinθ=−r·cosθ+ 1
r·sinθ+r·cosθ= 1
r(sinθ+cosθ) = 1
r=sinθ+^1 cosθExample C
Express the following equation using rectangular coordinates:r= 1 +2 cos^8 θ.
Solution:Use the fact thatr=±
√
x^2 +y^2 andrcosθ=x.r+ 2 r·cosθ= 8
±√
x^2 +y^2 + 2 x= 8
±√
x^2 +y^2 = 8 − 2 x
x^2 +y^2 = 64 − 32 x+ 4 x^2
− 3 x^2 + 32 x+y^2 − 64 = 0