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Polar Coordinates. Lesson 10.5. •. θ. r. Points on a Plane. (x, y). •. (r, θ ). Rectangular coordinate system Represent a point by two distances from the origin Horizontal dist, Vertical dist Also possible to represent different ways

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Presentation Transcript
points on a plane

θ

r

Points on a Plane

(x, y)

(r, θ)

  • 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
plot given polar coordinates
Plot Given Polar Coordinates
  • Locate the following
find polar coordinates
Find Polar Coordinates

• A

  • A =
  • B =
  • C =
  • D =

• B

• D

• C

What are the coordinates for the given points?

converting polar to rectangular
Converting Polar to Rectangular

r

y

θ

x

  • Given polar coordinates (r, θ)
    • Change to rectangular
  • By trigonometry
    • x = r cos θy = r sin θ
  • Try = ( ___, ___ )
converting rectangular to polar
Converting Rectangular to Polar

r

y

θ

x

  • 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 = ______
polar equations
Polar Equations

Note: for (r, θ)

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

θ in degrees

  • States a relationship between all the points (r, θ) that satisfy the equation
  • Example r = 4 sin θ
    • Resulting values
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
finding dy dx
Finding dy/dx
  • We know
    • r = f(θ) and y = r sin θ and x = r cos θ
  • Then
  • And
finding dy dx1
Finding dy/dx
  • Since
  • Then
example
Example
  • Given r = cos 3θ
    • Find the slope of the line tangent at (1/2, π/9)
    • dy/dx = ?
    • Evaluate

define for calculator
Define for Calculator

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

try this
Try This!
  • Find where the tangent line is horizontal for r = 2 cos θ
  • Find dy/dx
  • Set equal to 0, solve for θ
assignment
Assignment

Lesson 10.4

Page 736

Exercises 1 – 19 odd, 23 – 26 all

Exercises 69 – 91 EOO

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