# Polar Coordinates - PowerPoint PPT Presentation

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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

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

• Exampler = 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

• 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