Further Maths - Polar Coordinates

Created: November 09, 2021

Flashcards

What is the $y = x$ equivalent for polar coordinates?
\[r = \theta\]

What is $x$ in terms of $r$ and $\theta$ for polar coordinates?
\[x = r\cos\theta\]

What is $y$ in terms of $r$ and $\theta$ for polar coordinates?
\[y = r\sin\theta\]

What is $r$ in terms of $x$ and $y$ for polar coordinates?
\[r = \sqrt{x^2 + y^2}\]

What is $\theta$ in terms of $x$ and $y$?
\[\tan^{-1}\left(\frac{x}{y}\right)\]

2021-11-15

$r = a\cos n\theta$

\[r = a\cos\theta\]

What does this polar graph look like?

PHOTO ACOS1

\[r = a\cos\theta\]

How would you describe IN WORDS what this look like?

A circle along the $x$-axis starting at the origin and ending after a diameter $a$ long.

PHOTO ACOS1 What is the general polar equation for curves that look like this?
\[r = a\cos\theta\]
\[r = a\cos2\theta\]

What does this polar graph look like?

PHOTO ACOS2

PHOTO ACOS2 What is the general polar equation for curves that look like this?
\[r = a\cos2\theta\]
\[r = a\cos3\theta\]

What does this polar graph look like?

PHOTO ACOS3

PHOTO ACOS3 What is the general polar equation for curves that look like this?
\[r = a\cos3\theta\]
\[r = a\cos4\theta\]

What does this polar graph look like?

PHOTO ACOS4

PHOTO ACOS4 What is the general polar equation for curves that look like this?
\[r = a\cos4\theta\]
\[r = a\cos5\theta\]

What does this polar graph look like?

PHOTO ACOS5

PHOTO ACOS5 What is the general polar equation for curves that look like this?
\[r = a\cos5\theta\]

$r = a\sin n\theta$

\[r = a\sin\theta\]

What does this polar graph look like?

PHOTO ASIN1

\[r = a\sin\theta\]

How would you describe IN WORDS what this look like?

A circle along the $y$-axis starting at the origin and ending after a diameter $a$ long.

PHOTO ASIN1 What is the general polar equation for curves that look like this?
\[r = a\sin\theta\]
\[r = a\sin2\theta\]

What does this polar graph look like?

PHOTO ASIN2

PHOTO ASIN2 What is the general polar equation for curves that look like this?
\[r = a\sin2\theta\]
\[r = a\sin3\theta\]

What does this polar graph look like?

PHOTO ASIN3

PHOTO ASIN3 What is the general polar equation for curves that look like this?
\[r = a\sin3\theta\]
\[r = a\sin4\theta\]

What does this polar graph look like?

PHOTO ASIN4

PHOTO ASIN4 What is the general polar equation for curves that look like this?
\[r = a\sin4\theta\]
\[r = a\sin5\theta\]

What does this polar graph look like?

PHOTO ASIN5

PHOTO ASIN5 What is the general polar equation for curves that look like this?
\[r = a\sin5\theta\]

PHOTO ASIN2 What do the dashed lines represent here?

Where the polar equation gives negative results.

PHOTO ANGLE MAX SIN 2 Where would you expect the maximum “bump” to start for a polar equation $r = a\sin n\theta$?
\[\theta = \frac{\pi}{2n}\]

Cardioids

PHOTO ABCOS POINT What is the name for shapes like these?

Cardioids.

\[r = a + b\cos\theta\]

What does this polar graph look like, for $a = \vert b \vert $?

PHOTO ABCOS POINT

\[r = a + b\cos\theta\]

What does this polar graph look like, for $a > \vert b \vert $?

PHOTO ABCOS BUMP

\[r = a + b\sin\theta\]

What does this polar graph look like, for $a = \vert b \vert $?

PHOTO ABSIN POINT

\[r = a + b\sin\theta\]

What does this polar graph look like, for $a > \vert b \vert $?

PHOTO ABSIN BUMP

Integration

Given a polar equation

\[r = ...\]

what is the formula for the area between angles $\alpha$ and $\beta$?
\[\frac{1}{2} \int^\alpha_\beta r^2 d\theta\]

Where does the polar integration formula

\[\frac{1}{2} \int^\alpha _ \beta r^2 d\theta\]

come from?

The formula for arc area, $\frac{1}{2}r^2\theta$

PHOTO ACOS3 Why do you have to be careful picking limits to find the area of one loop of this curve?

Because you would’ve thought you could pick $\pi/2$ and $-\pi/2$ but you actually have to use the closest tangent so you don’t include unnecessary area.

2021-11-17

\[x = r\cos\theta\] \[y = r\sin\theta\]

Given that $r = \cos\theta$ what is the parametric form of the polar equation with parameter $\theta$?
\[(r\cos^2\theta, r\cos\theta\sin\theta)\]

For

\[r = f(\theta)\]

what is the formula for $x$?
\[x = f(\theta)\cos\theta\]

For

\[r = f(\theta)\]

what is the formula for $y$?
\[y = f(\theta)\sin\theta\]

If

\[\frac{\text{d}x}{\text{d}\theta} = 0\]

what is true about a polar curve for that value of $\theta$?

It is perpendicular to the initial line ($\theta = 0$).

If

\[\frac{\text{d}y}{\text{d}\theta} = 0\]

what is true about a polar curve for that value of $\theta$?

It is parallel to the initial line ($\theta = 0$).

What would you set equal to $0$ to find the values of $\theta$ for which a polar curve is parallel to the initial line?
\[\frac{\text{d}y}{\text{d}\theta} = 0\]

What would you set equal to $0$ to find the values of $\theta$ for which a polar curve is perpendicular to the initial line?
\[\frac{\text{d}x}{\text{d}\theta} = 0\]

If a value of $\theta = \frac{\pi}{2}$ gives a value of $r = 1$ what is the coordinate?
\[\left(1, \frac{\pi}{2}\right)\]


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