Q 25. - Calculus

Question

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

$\left(1,0\right)$

Step-by-Step Solution

Answer

The polar coordinates are $\left(-1,\left(2\mathrm{k}+1\right)\mathrm{\pi }\right) \mathrm{and} \left(1,2\mathrm{k\pi }\right)$, for any integer k.

Step 1. Given information.

The given rectangular coordinate is:

 $\left(1,0\right)$

Here $x=1 \mathrm{and} y=0$.

Step 2. Find the value of r .

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

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{{\left(1\right)}^2+{\left(0\right)}^2} \\ r=\sqrt{1+0} \\ r=\pm 1 \end{gathered}$$

Step 3. Find the value of θ .

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

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

$\theta =\mathrm{\pi }, 2\mathrm{\pi } \left( \mathrm{Since} \mathrm{tan\pi }=0; \tan 2\mathrm{\pi }=0\right)$

Step 4. Find the polar coordinates.

Now take $r=-1, \theta =\mathrm{\pi }$,

Then $\left(\mathrm{r},\mathrm{\theta }\right)=\left(-1,\mathrm{\pi }\right)$

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right) \mathrm{are} \left(-1,\left(2\mathrm{k}+1\right)\mathrm{\pi }\right)$ for any integer k.

Now take $\mathrm{r}=1, \mathrm{\theta }=2\mathrm{\pi }$,

Then $\left(\mathrm{r},\mathrm{\theta }\right)=\left(1,2\mathrm{\pi }\right)$.

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right) \mathrm{are} \left(1,2\mathrm{k\pi }\right)$, for any integer k.

Therefore, all the polar coordinates are $\left(-1,\left(2\mathrm{k}+1\right)\mathrm{\pi }\right) \mathrm{and} \left(1,2\mathrm{k\pi }\right)$,

for any integer k.