Q 26. - Calculus

Question

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

$\left(0,-3\right)$

Step-by-Step Solution

Answer

The polar coordinates are $\left(-3,-\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right) \mathrm{and} \left(3,\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$, for any integer k. 

Step 1. Given information.

The given rectangular coordinate is:

 $\left(0,-3\right)$

Here $x=0 \mathrm{and} y=-3$

Step 2. Find the value of r .

To find the value of r, use the formula $r^2=x^2+y^2$,

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{{\left(0\right)}^2+{\left(-3\right)}^2} \\ r=\sqrt{0+9} \\ r=\pm 3 \end{gathered}$$

Step 3. Find the value of θ .

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

$$\begin{gathered} \tan \theta =\frac{y}{x} \\ \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ \theta ={\tan }^{-1}\left(\frac{-3}{0}\right) \\ \theta ={\tan }^{-1}\left(\infty \right) \end{gathered}$$

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

Step 4. Find the polar coordinates.

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

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right) \mathrm{are} \left(-3,-\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$ for any integer k.

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

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right) \mathrm{are} \left(3,\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$, for any integer k.

Therefore, all the polar coordinates are $\left(-3,-\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right) \mathrm{and} \left(3,\frac{\mathrm{\pi }}{2}+2\mathrm{k\pi }\right)$,

for any integer k.