Polar Coordinates

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Polar Coordinates

• Lesson 10.5

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Points on a Plane

• Rectangular coordinate system
• Represent a point by two distances from the
  origin
• Horizontal dist, Vertical dist
• Also possible to represent different ways
• Consider using dist from origin, angle formed
  with positive x-axis

(x, y)

(r, ?)
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Plot Given Polar Coordinates

• Locate the following

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Find Polar Coordinates

• What are the coordinates for the given points?

A

• A
• B
• C
• D

B
D
C
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Converting Polar to Rectangular

• Given polar coordinates (r, ?)
• Change to rectangular
• By trigonometry
• x r cos ?y r sin ?
• Try ( ___, ___ )

r
y
?
x
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Converting Rectangular to Polar

• Given a point (x, y)
• Convert to (r, ?)
• By Pythagorean theorem r2 x2 y2
• By trigonometry
• Try this one for (2, 1)
• r ______
• ? ______

r
y
?
x
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Polar Equations

• States a relationship between all the points (r,
  ?) that satisfy the equation
• Example r 4 sin ?
• Resulting values

Note for (r, ?) It is ? (the 2nd element that
is the independent variable
? in degrees
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Graphing Polar Equations

• Set Mode on TI calculator
• Mode, then Graph gt Polar
• Note difference of Y screen

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Graphing Polar Equations

• Also best to keepangles in radians
• Enter function in Y screen

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Graphing Polar Equations

• Set Zoom to Standard,
• then Square

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Try These!

• For r A cos B ?
• Try to determine what affect A and B have
• r 3 sin 2?
• r 4 cos 3?
• r 2 5 sin 4?

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Finding dy/dx

• We know
• r f(?) and y r sin ? and x r cos ?
• Then
• And

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Finding dy/dx

• Since
• Then

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Example

• Given r cos 3?
• Find the slope of the line tangent at (1/2, p/9)
• dy/dx ?
• Evaluate

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Define for Calculator

• It is possible to define this derivative as a
  function on your calculator

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Try This!

• Find where the tangent line is horizontal for r
  2 cos ?
• Find dy/dx
• Set equal to 0, solve for ?

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Assignment

• Lesson 10.4
• Page 736
• Exercises 1 19 odd, 23 26 all
• Exercises 69 91 EOO