Q 27. - Calculus

Question

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.

$\left(0,2\right)$

Step-by-Step Solution

Answer

The polar coordinates are $\left(-2,-\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right) \mathrm{and} \left(2,\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$, for any integer k.  

Step 1. Given information.

The given rectangular coordinate is:

$\left(0,2\right)$

Here $x=0 \mathrm{and} y=2$.

Step 2. Find the value of r .

To find the value of r, use the formula $r=\sqrt{x^2+y^2}$:

$$\begin{gathered} r=\sqrt{{\left(0\right)}^2+{\left(2\right)}^2} \\ r=\sqrt{0+4} \\ r=\pm 2 \end{gathered}$$

Step 3. Find the value of θ .

Use the formula $\tan \theta =\frac{y}{x}$,

$$\begin{gathered} \tan \theta =\frac{y}{x} \mathrm{then} \theta = {\tan }^{-1}\left(\frac{y}{x}\right) \\ \theta = {\tan }^{-1}\left(\frac{2}{0}\right) \\ \theta = {\tan }^{-1}\left(\infty \right) \end{gathered}$$

$\theta =-\frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2} \left[\mathrm{Since} \tan \frac{\mathrm{\pi }}{2}=\infty ; \tan \frac{3\mathrm{\pi }}{2}=\infty \right]$

Step 4. Find the polar coordinates.

Take, $r=-2, \theta =-\frac{\mathrm{\pi }}{2} \mathrm{then} \left(r,\theta \right)=\left(-2,-\frac{\mathrm{\pi }}{2}\right)$

The coordinates of $\left(r,\theta \right) \mathrm{are} \left(-2,-\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$, for any integer k.

Now take $\mathrm{r}=2, \mathrm{\theta }=\frac{3\mathrm{\pi }}{2} \mathrm{then} \left(\mathrm{r},\mathrm{\theta }\right)=\left(2,\frac{3\mathrm{\pi }}{2}\right)$,

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right) \mathrm{are} \left(2,\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$, for any integer k.

Therefore, all the polar coordinates are $\left(-2,-\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right) \mathrm{and} \left(2,\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$, 

for any integer k.