Section 11.3 Polar Coordinates. POLAR COORDINATES The polar coordinate system is another way to...
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Transcript of Section 11.3 Polar Coordinates. POLAR COORDINATES The polar coordinate system is another way to...
Section 11.3
Polar Coordinates
POLAR COORDINATES
The polar coordinate system is another way to specify points in a plane. Points are specified by the directed distance, r, form the pole and the directed angle, θ, measures counter-clockwise from the polar axis. The pole has coordinates (0, θ).
UNIQUESNESS OF POLAR COORDINATES
In polar coordinates, ordered pairs of points are NOT unique; that is, there are many “names” to describe the same physical location.
The point (r, θ) can also be represented by (r, θ + 2kπ) and (− r, θ + [2k + 1]π).
CONVERTING BETWEEN RECTANGULAR AND POLAR
COORDINATES• Polar coordinates to
rectangular coordinates
• Rectangular coordinates to polar coordinates
sin;cos ryrx
x
yyxr tan;222
FUNCTIONS IN POLAR COORDINATES
A function in polar coordinates has the formr = f (θ).
Some examples:r = 4cos θr = 3r = −3sec θ
POLAR EQUATIONS TO RECTANGULAR EQUATIONS
22:NOTE yxr
To convert polar equations into rectangular equations use:
222
2222;sin;cos yxr
yx
y
yx
x
RECTANGULAR EQUATIONS TO POLAR EQUATIONS
22:NOTE yxr
To convert rectangular equations to polar equations use:
x
yry
yxrrx
tansin
cos 222
HORIZONTAL AND VERTICAL LINES
1. The graph of r sin θ = a is a horizontal line a units above the pole if a is positive and |a| units below the pole if a is negative.
2. The graph of r cos θ = a is a vertical line a units to the right of the pole if a is positive and |a| units to the left of the pole if a is negative.
POLAR EQUATIONS OF CIRCLES
• The equation r = a is a circle of radius |a| centered at the pole.
• The equation r = acos θ is a circle of radius |a/2|, passing through the pole, and with center on θ = 0 or θ = π.
• The equation r = asin θ is a circle of radius |a/2|, passing through the pole, and with center on θ = π/2 or θ = 3π/2.
ROSE CURVES
• The rose curve has 2a leaves (petals) if a is an even number.
• The rose curve has a leaves (petals) if a is an odd number.
• The leaves (petals) have length b.• To graph rose curves pick multiples of (π/2) · (1/a)
The equationsr = bsin(aθ)r = bcos(aθ)
both have graphs that are called rose curves.
LIMAÇONS
• If |a/b| < 1, then the limaçon has an inner loop. For example: r = 3 − 4cos θ.
• If |a/b| = 1, then the limaçon is a “heart-shaped” graph called a cardiod. For example: r = 3 + 3sin θ.
The graphs of the equationsr = a ± bsin θr = a ± bcos θ
are called limaçons.
LIMAÇONS (CONTINUED)
• If 1 < |a/b| < 2, then the limaçon is dimpled. For example: r = 3 + 2cos θ.
• If |a/b| ≥ 2, then the limaçon is convex. For example: r = 3 − sin θ.
TANGENTS TO POLAR CURVES
Given a polar curve r = f (θ), the Cartesian coordinates of a point on the curve are:
x = r cos θ = f (θ) cos θ
y = r sin θ = f (θ) sin θ
Hence,
sincos
cossin
rddr
rddr
ddxddy
dx
dy