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??
$$r = a\cos\theta$$ 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.
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??
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??
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??
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??
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??
$$r = a\sin\theta$$ 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.
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??
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??
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??
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??
What is the general polar equation for curves that look like this??
$$ r = a\sin5\theta $$
What do the dashed lines represent here??
Where the polar equation gives negative results.
Where would you expect the maximum “bump” to start for a polar equation $r = a\sin n\theta$??
$$ \theta = \frac{\pi}{2n} $$
Cardioids
What is the name for shapes like these??
Cardioids.
$$r = a + b\cos\theta$$ What does this polar graph look like, for $a = |b|$??
$$r = a + b\cos\theta$$ What does this polar graph look like, for $a > |b|$??
$$r = a + b\sin\theta$$ What does this polar graph look like, for $a = |b|$??
$$r = a + b\sin\theta$$ What does this polar graph look like, for $a > |b|$??
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, $\frac{1}{2}r^2\theta$
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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Metadata
date: 2021-11-09 15:36
tags:
- '@?public'
- '@?school'
- '@?further-maths'
- '@?polar'
- '@?year-2'
title: Further Maths - Polar Coordinates
