6.3 Polar Coordinates Day One

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6.3 Polar CoordinatesDay One

• Objectives
• Plot points in the polar coordinate system
• Find multiple sets of polar coordinate for a
  given point
• Convert a point from polar to rectangular
  coordinates
• Convert a point from rectangular to polar
  coordinates
• Pg. 655 2-48 (even)

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Defining points in the polar system

• Location of a point is based on radius (distance
  from the origin) and theta (the angle the radius
  moves from standard position (positive x-axis in
  a cartesian system)
• (r, T) is the point in polar coordinates.
• r and T can be positive, negative, or zero.

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The point described in polar coordinates by (2,
3p/4) would look like this                     
                                     
1. Plot the point (3, 315o) on the polar graph
provided.
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2. Plot the point (2, 60)
3. Plot the point (4, 165)
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4. Plot the point (-2, p)
5. Plot the point (-1, )
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Multiple Representations of Points If n is an
integer, then the point (r, T) can be represented
as (r, T 2np) or (-r, T p 2np)

• Find another representation of (5, ) in
  which
• r is positive and 2p lt T lt 4p
• r is negative and 0 lt T lt 2p
• r is positive and - 2p lt T lt 0

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Converting from Polar Coordinates to Rectangular
Coordinates
If cos ? x/r then x r cos ?
If sin ? y/r then y r sin ?

• Find the rectangular coordinates for the points
  given in polar coordinates.
• (3, p) 10. (-10, )

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Converting from Rectangular Coordinates to Polar
Coordinates
From Pythagoras, we have r2 x2 y2 and,
thus, and basic trigonometry gives us
             
Step 1 Determine the quadrant in which the
point resides. Step 2 Find r by computing the
distance from the origin to (x,y) using Step 3
Find T using .
Make sure the angles fits the quadrant
determined in step 1. Note We are assuming
that r gt 0 and 0 lt T lt 2p
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11. Find the polar coordinates of the point
whose rectangular coordinates are (1, - )
12. Find the polar coordinates of the point
whose rectangular coordinates are (0, -4)