Intro to polar coordinates
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Intro to Polar Coordinates. •. θ. r. 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). •.

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Intro to Polar Coordinates

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Intro to polar coordinates

Intro to Polar Coordinates


Points on a plane

θ

r

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, θ)


Plot given polar coordinates

Plot Given Polar Coordinates

  • Locate the following


Find polar coordinates

Find Polar Coordinates

  • What are the coordinates for the given points?

• A

  • A =

  • B =

  • C =

  • D =

• B

• D

• C


Converting polar to rectangular

Converting Polar to Rectangular

  • Given polar coordinates (r, θ)

    • Change to rectangular

  • By trigonometry

    • x = r cos θy = r sin θ

  • Try = ( ___, ___ )

r

y

θ

x


Converting rectangular to polar

Converting Rectangular to Polar

  • Given a point (x, y)

    • Convert to (r, θ)

  • By Pythagorean theorem r2 = x2 + y2

  • By trigonometry

  • Try this one … for (2, 1)

    • r = ______

    • θ = ______

r

y

θ

x


Polar equations

Polar Equations

  • States a relationship between all the points (r, θ) that satisfy the equation

  • Exampler = 4 sin θ

    • Resulting values

Note: for (r, θ)

It is θ (the 2nd element that is the independent variable

θ in degrees


Graphing polar equations

Graphing Polar Equations

  • Set Mode on TI calculator

    • Mode, then Graph => Polar

  • Note difference of Y= screen


Graphing polar equations1

Graphing Polar Equations

  • Also best to keepangles in radians

  • Enter function in Y= screen


Graphing polar equations2

Graphing Polar Equations

  • Set Zoom to Standard,

    • then Square


Try these

Try These!

  • For r = A cos Bθ

    • Try to determine what affect A and B have

  • r = 3 sin 2θ

  • r = 4 cos 3θ

  • r = 2 + 5 sin 4θ

Experiment with Polar Function Spreadsheet


Write polar equation in rectangular form

Write Polar Equation in Rectangular Form

  • Given r = 2 sin θ

    • Write as rectangular equation

  • Use definitions

    • And identitiesGraph the given equation for clues


Write polar equation in rectangular form1

Write Polar Equation in Rectangular Form

  • Given r = 2 sin θ

    • We know

    • Thus

    • And


Write rectangular equation in polar form

Write Rectangular Equation in Polar Form

  • Consider 2x – 3y = 6

    • As before, usedefinitions


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