Post on 21-Jan-2021
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Polar Coordinates
Section 11.4
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Polar Coordinates • Polar coordinates is a new system for assigning coordinates
to points in the plane. • We start with an origin point, called the pole, and a ray
called the polar axis. • We locate a point P using two coordinates, (r,Θ)• r represents a directed distance from the pole • Θ is a measure of rotation from the polar axis.
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Example • We wish to plot the point P with polar coordinates (4,
5π/6)• We start at the pole, move out along the polar axis 4
units, then rotate 5π/6 radians counterclockwise
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Example (continued)• We may also visualize this process by thinking of the
rotation first• To plot (4, 5π/6) we rotate 5π/6 counterclockwise from
the polar axis, then move outwards from the pole 4 units
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Case r < 0• To plot (3.5, π/4) we rotate π/4 counterclockwise from
the polar axis, then move back through the pole 3.5 units
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Case Θ < 0• Θ < 0 means the rotation away from the polar axis
(clockwise instead of counterclockwise)• To plot (3.5, 3π/4) we rotate 3π/4 clockwise
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Point Representation • In Cartesian coordinates (a, b) and (c, d) represent the
same point if and only if a = c and b = d• In Polar Coordinates a point can be represented by
infinitely many polar coordinate pairs
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Example • For each point in polar coordinates given below plot the
point and then give two additional expressions for the point, one of which has r > 0 and the other with r < 0
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Equivalent Representations of Points in Polar Coordinates
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Conversion Between Rectangular and Polar Coordinates
• Suppose P is represented in rectangular coordinates as (x, y) and in polar coordinates as (r, Θ).
• Then•
• • •
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Example • Convert each point in rectangular coordinates given
below into polar coordinates with r ≥ 0 and 0 ≤ Θ< 2π. • Use exact values if possible and round any approximate
values to two decimal places. • Check your answer by converting them back to
rectangular coordinates
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Converting equations from one system to another
• Just as we've used equations in x and y to represent relations in rectangular coordinates, equations in the variables r and Θ represent relations in polar coordinates.
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Example • Convert each equation in rectangular coordinates into
an equation in polar coordinates
• Convert each equation in polar coordinates into an equation in rectangular coordinates
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