Polar coordinates
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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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Polar Coordinates

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Polar coordinates

Polar Coordinates

Lesson 10.5


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

  • Exampler = 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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