Lecture 32: Polar Coordinates (11.3)
Instead of using (x, y), we describe a point by
(r, θ) in the polar coordinates where r is its distance from the origin and θ is the angle it makes
with the positive x−axis.
The Cartesian Coordinates of a point (x, y) and
its polar coordinates (r, θ) are related by the
equations
x = r cos θ , y = r sin θ.
r 2 = x2 + y 2 ,
y
tan θ = , x 6= 0.
x
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Unlike in the rectangular coordinates where each
point (x, y) is uniquely defined, a point in polar
coordinates can have infinitely many description.
For example, (r, θ) = (1, 0) = (1, 2nπ) . In fact,
(r, θ) = (r, θ ± 2nπ) = (−r, θ ± (2n + 1)π)
Negative r : By convention, r > 0, ”−r”
means swing to the other side of angle θ. (−r, θ)
is the reflection of (r, θ) through the origin. (−r, θ)
and (r, θ + π) represent the same point.
The pole(the origin) is denoted by (0, θ) for any
angle θ.
ex. Plot the points whose polar coordinates are
(r, θ) = (1, 0), (1, 2π), (1, −4π), (0, π), (0, π/2), (0, 7.4π),
(−1, − π2 ), (−1, π2 ), (4, π)(−4, 0), (2, − π3 )
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ex. Find 2 other pairs of polar coordinates of
the point (−1, −π/2), one with r > 0, and one
with r < 0.
ex. Convert the point (r, θ) = (3, 2π
3 ) from polar
coordinates to Cartesian Coordinates.
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√
ex. Convert the point (x, y) = (−1, − 3) from
Cartesian Coordinates to polar coordinates.
ex. Find a polar equation for the curve represented by the given Cartesian equation.
x2 + y 2 = 4
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ex. Graph the polar curves by finding a Cartesian equation for the curve:
1. r = 2
2. r = 3 cos θ
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ex. Construct a polar equation for a circle of
radius a with its center at x = a, y = 0.
Circle family : r = a, r = 2a sin θ, r = 2a cos θ.
r = a is a circle centered at the origin. Others
are ’off-centered circles. The choice of cos θ or
sin θ determines if the circle sits on the x− or
y−axis.
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Symmetry in Polar graphs:
1. (r, θ) and (r, −θ) are symmetric with respect
to the x−axis.
2. (r, θ) and (−r, θ) are symmetric with respect
to the pole.
3. (r, θ) and (r, π − θ) are symmetric to the y−
axis.
ex. Graph the polar curve r = cos 2θ.
Method 1: Plot points:
Since cosine is an even function, cos 2θ = cos 2(−θ);
(r, θ) and (r, −θ) are both on the curve. The
curve is symmetric w.r.t. the x−axis. Using the
symmetry, it suffices to plot points for 0 ≤ θ ≤
π.
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Method 2: Analyze r as a function of θ in rectangular coordinates. ( to be continued · · ·)
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