Further Maths - Polar Coordinates

Pearson Edexcel Further Mathematics 2022

Created: November 09, 2021 | Read markdown | About these notes | View in context | Study these flashcards |

Flashcards

Cartesian and polar conversions

What is the $y = x$ equivalent for polar coordinates?

polar-equivalent-of-y-equals-x
\[r = \theta\]

What is $x$ in terms of $r$ and $\theta$ for polar coordinates?

x-from-polar
\[x = r\cos\theta\]

What is $y$ in terms of $r$ and $\theta$ for polar coordinates?

y-from-polar
\[y = r\sin\theta\]

What is $r$ in terms of $x$ and $y$ for polar coordinates?

r-from-cartesian
\[r = \sqrt{x^2 + y^2}\]

What is $\theta$ in terms of $x$ and $y$?

theta-from-cartesian
\[\tan^{-1}\left(\frac{y}{x}\right)\]

$r = a\cos n\theta$

\[r = a\cos\theta\]

What does this polar graph look like?

acos-theta-shape
\[r = a\cos\theta\]

How would you describe IN WORDS what this look like?

acos-theta-description

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

What is the general polar equation for curves that look like this?

acos-theta-equation-from-shape
\[r = a\cos\theta\]
\[r = a\cos2\theta\]

What does this polar graph look like?

acos-2theta-shape

What is the general polar equation for curves that look like this?

acos-2theta-equation-from-shape
\[r = a\cos2\theta\]
\[r = a\cos3\theta\]

What does this polar graph look like?

acos-3theta-shape

What is the general polar equation for curves that look like this?

acos-3theta-equation-from-shape
\[r = a\cos3\theta\]
\[r = a\cos4\theta\]

What does this polar graph look like?

acos-4theta-shape

What is the general polar equation for curves that look like this?

acos-4theta-equation-from-shape
\[r = a\cos4\theta\]
\[r = a\cos5\theta\]

What does this polar graph look like?

acos-5theta-shape

What is the general polar equation for curves that look like this?

acos-5theta-equation-from-shape
\[r = a\cos5\theta\]

$r = a\sin n\theta$

\[r = a\sin\theta\]

What does this polar graph look like?

asin-theta-shape
\[r = a\sin\theta\]

How would you describe IN WORDS what this look like?

asin-theta-description

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

What is the general polar equation for curves that look like this?

asin-theta-equation-from-shape
\[r = a\sin\theta\]
\[r = a\sin2\theta\]

What does this polar graph look like?

asin-2theta-shape

What is the general polar equation for curves that look like this?

asin-2theta-equation-from-shape
\[r = a\sin2\theta\]
\[r = a\sin3\theta\]

What does this polar graph look like?

asin-3theta-shape

What is the general polar equation for curves that look like this?

asin-3theta-equation-from-shape
\[r = a\sin3\theta\]
\[r = a\sin4\theta\]

What does this polar graph look like?

asin-4theta-shape

What is the general polar equation for curves that look like this?

asin-4theta-equation-from-shape
\[r = a\sin4\theta\]
\[r = a\sin5\theta\]

What does this polar graph look like?

asin-5theta-shape

What is the general polar equation for curves that look like this?

asin-5theta-equation-from-shape
\[r = a\sin5\theta\]

What do the dashed lines represent here?

dashed-lines-negative-r

Where the polar equation gives negative results.

Where would you expect the maximum “bump” to start for a polar equation $r = a\sin n\theta$?

asin-petal-start-angle
\[\theta = \frac{\pi}{2n}\]

Cardioids

What is the name for shapes like these?

cardioid-name

Cardioids.

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

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

cardioid-cos-equal
\[r = a + b\cos\theta\]

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

cardioid-cos-greater
\[r = a + b\sin\theta\]

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

cardioid-sin-equal
\[r = a + b\sin\theta\]

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

cardioid-sin-greater

Area by integration

Given a polar equation

\[r = ...\]

what is the formula for the area between angles $\alpha$ and $\beta$?

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

Where does the polar integration formula

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

come from?

polar-area-formula-origin

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

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

polar-area-limit-choice

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.

Tangents and parametric form

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

Given that $r = \cos\theta$ what is the parametric form of the polar equation with parameter $\theta$?

polar-to-parametric-example
\[(r\cos^2\theta, r\cos\theta\sin\theta)\]

For

\[r = f(\theta)\]

what is the formula for $x$?

x-from-polar-function
\[x = f(\theta)\cos\theta\]

For

\[r = f(\theta)\]

what is the formula for $y$?

y-from-polar-function
\[y = f(\theta)\sin\theta\]

If

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

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

dx-dtheta-zero-perpendicular

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 $\theta$?

dy-dtheta-zero-parallel

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?

find-parallel-tangents
\[\frac{\text{d}y}{\text{d}\theta} = 0\]

What would you set equal to $0$ to find the values of $\theta$ for which a polar curve is perpendicular to the initial line?

find-perpendicular-tangents
\[\frac{\text{d}x}{\text{d}\theta} = 0\]

If a value of $\theta = \frac{\pi}{2}$ gives a value of $r = 1$ what is the coordinate?

polar-coordinate-r-theta-order
\[\left(1, \frac{\pi}{2}\right)\]


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