# Further Maths - Polar Coordinates

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/polar-graphs/ · Updated: 2021-11-09 · Tags: school, further-maths, polar, year-2

## 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](acos1.png)

##### $$r = a\cos\theta$$ 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](acos1.png) 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](acos2.png)

##### ![PHOTO ACOS2](acos2.png) 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](acos3.png)

##### ![PHOTO ACOS3](acos3.png) 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](acos4.png)

##### ![PHOTO ACOS4](acos4.png) 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](acos5.png)

##### ![PHOTO ACOS5](acos5.png) 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](asin1.png)

##### $$r = a\sin\theta$$ 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](asin1.png) 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](asin2.png)

##### ![PHOTO ASIN2](asin2.png) 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](asin3.png)

##### ![PHOTO ASIN3](asin3.png) 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](asin4.png)

##### ![PHOTO ASIN4](asin4.png) 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](asin5.png)

##### ![PHOTO ASIN5](asin5.png) What is the general polar equation for curves that look like this??
$$
r = a\sin5\theta
$$

##### ![PHOTO ASIN2](asin2.png) What do the dashed lines represent here??
Where the polar equation gives negative results.

##### ![PHOTO ANGLE MAX SIN 2](angle-max-sin2.png) 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](abcos-point.png) What is the name for shapes like these??
Cardioids.

##### $$r = a + b\cos\theta$$ What does this polar graph look like, for $a = |b|$??
![PHOTO ABCOS POINT](abcos-point.png)

##### $$r = a + b\cos\theta$$ What does this polar graph look like, for $a > |b|$??
![PHOTO ABCOS BUMP](abcos-bump.png)

##### $$r = a + b\sin\theta$$ What does this polar graph look like, for $a = |b|$??
![PHOTO ABSIN POINT](absin-point.png)

##### $$r = a + b\sin\theta$$ What does this polar graph look like, for $a > |b|$??
![PHOTO ABSIN BUMP](absin-bump.png)

#### 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](acos3.png) 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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