AP Calculus BC - Polar Equations

√x^2 + y^2

(Cartesian to polar) r =

arctan y/x

(Cartesian to polar) θ =

r cos θ

(Polar to cartesian) x =

r sin θ

(Polar to cartesian) y =

Circles, Roses, lines, and limaçons

Types of polar equations

r = a
r = a sin θ
r = a cos θ

Equations of circles

The x-axis

A polar equation that has cos θ has symmetry with...

The y-axis

A polar equation that has sin θ has symmetry with...

The length of the diameter

In a circle equation (excluding r = a), a is...

The location of the circle in the polar plane

In a circle equation (excluding r = a), the sign of a determines:

0 ≤ θ ≤ 2π

Range of θ for r = a

0 ≤ θ ≤ π

Range of θ for r = a sin θ & r = a cos θ

r = a sin nθ
r = a cos nθ

Types of rose equations:

The length of the petals

In a rose equation (r = a sin nθ & r = a cos nθ), a is equal to:

The number of petals the graph will have

In a rose equation (r = a sin nθ & r = a cos nθ), if n is odd, then n is equal to:

n times 2 is equal the number of petals the rose will have (2n).

In a rose equation (r = a sin nθ & r = a cos nθ), if n is even, then n is equal to:

0 ≤ θ ≤ 2π

Range of rose equations (r = a sin nθ & r = a cos nθ):

r = a ± b sin θ
r = a ± b cos θ

Equations of limaçons:

Cardiods, Iner Loops, and Beans

Types of limaçons:

0 ≤ θ ≤ 2π

Ranges of limaçons:

Where the graph is located

In a limaçon, the sign of b determines:

heart

A cardiod graph roughly has the shape of a

a = b

For a limaçon to be a cardiod, what is the relationship between a & b:

The length from the origin to the intercepts it has with either axis (the axis which is being intercepted depends on wether it is cos θ or sin θ)

In a cardiod, a is equal to:

The point from the origin to the max length

In a cardiod, a + b determines:

Where most of the graph is located.

In a cardiod, the sign of b determines:

0

In a cardiod, b-a is equal to:

It has an inner loop that then loops around and connects to make a bigger shape

What does an inner loop graph look like?

a < b

For a limaçon to be an inner loop, what is the relationship between a & b:

The length of the intercepts the graph will have

In an inner loop, what does the value of a determine?

Where most of the graph will be located in

In an inner loop, what does the sign of b determine?

The vertex of the inner loop

In an inner loop, what does b-a determine?

The vertex of the big shape

In an innner loop, what does b+a determine?

Like a bean

What does the graph of a bean limaçon look like?

a > b

For a limaçon to be a bean, what is the relationship between a & b:

The length the short vertext

In a bean, what does a determine?

Where most of the graph will be

In a bean, what does the sign of b determine?

The length of the main long vertext

In a bean, what does the value of a+b determine?

The length of the main short vertext

In a bean, what does the vale of b-a determine?

A line

What does a line graph look like?

r = a sec θ
r = a csc θ

What are the equation for line graphs?

Vertical line at the value of a

r = a sec θ is a

Horizontal line at the value of a

r = a csc θ is a

Trig Idenitity: cos^2 θ=

1/2(1+ cos 2θ)

Trig Idenitity: sin^2 θ=

1/2(1-cos 2θ)

At what values does a polar equation have horizontal tangent lines?

When dy/dθ = 0 and dx/dθ ≠ 0

How can we find a horizontal tangent line?

We must find values that make dy/dθ = 0 and then substitute them into dx/dθ to ensure that dx/dθ ≠ 0 at that value.

At what values does a polar equation have vertical tangent lines?

When dx/dθ = 0 and dy/dθ ≠ 0

How can we find a vertical tangent line?

We must find values that make dx/dθ = 0 and then substitute them into dx/dθ to ensure that dy/dθ ≠ 0 at that value.

What is dy/dx in a Polar Equation?

dy/dθ / dx/dθ

What is dy/dθ / dx/dθ?

r cos θ + r' sin θ
-------------------------------
r(-sin θ) + r' (cos θ)

What is dy/dθ ?

r cos θ + r' sin θ

What is dx/dθ ?

r(-sin θ) + r' (cos θ)