Polar Coordinates

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# Polar Coordinates - PowerPoint PPT Presentation

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

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
• Consider using dist from origin, angle formed with positive x-axis
Plot Given Polar Coordinates
• Locate the following
Find Polar Coordinates

• A

• A =
• B =
• C =
• D =

• B

• D

• C

What are the coordinates for the given points?

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

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

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
• Set Mode on TI calculator
• Mode, then Graph => Polar
• Note difference of Y= screen
Graphing Polar Equations

Also best to keepangles in radians

Enter function in Y= screen

Graphing Polar Equations
• Set Zoom to Standard,
• then Square
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
• We know
• r = f(θ) and y = r sin θ and x = r cos θ
• Then
• And
Finding dy/dx
• Since
• Then
Example
• Given r = cos 3θ
• Find the slope of the line tangent at (1/2, π/9)
• dy/dx = ?
• Evaluate

Define for Calculator

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

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

Lesson 10.4

Page 736

Exercises 1 – 19 odd, 23 – 26 all

Exercises 69 – 91 EOO