Question

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

$\left(3,4\right)$

Short answer

The polar coordinates are $\left(-5,0.927+2\mathrm{k\pi }\right)\mathrm{and}\left(5,0.927+2\mathrm{k\pi }\right)$, for any integer k.

Step-by-step solution

Step 1. Given information.

The given rectangular coordinate is:

$\left(3,4\right)$

Here$x=3\mathrm{and}y=4$.

Step 2. Find the value of r.

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

$$\begin{gathered} r=\sqrt{{\left(3\right)}^2+{\left(4\right)}^2} \\ r=\sqrt{9+16} \\ r=\sqrt{25} \\ r=\pm 5 \end{gathered}$$

Step 3. Find the value of θ.

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

$$\begin{gathered} \tan \theta =\frac{y}{x}\mathrm{then}\mathrm{\theta }={\tan }^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right) \\ \mathrm{\theta }={\tan }^{-1}\left(\frac{4}{3}\right) \\ \mathrm{\theta }=0.927 \end{gathered}$$

Step 4. Find the polar coordinates.

Take $r=-5,\mathrm{\theta }=0.927;\mathrm{then}\left(\mathrm{r},\mathrm{\theta }\right)=\left(-5,0.927\right)$,

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right)\mathrm{are}\left(-5,0.927+2\mathrm{k\pi }\right)$,for any integer k.

Now take $\mathrm{r}=5,\mathrm{\theta }=0.927;\mathrm{then}\left(\mathrm{r},\mathrm{\theta }\right)=\left(5,0.927\right)$,

The coordinates of $\left(\mathrm{r},\mathrm{\theta }\right)\mathrm{are}\left(5,0.927+2\mathrm{k\pi }\right)$, for any integer k.

Therefore, all the polar coordinates are $\left(-5,0.927+2\mathrm{k\pi }\right)\mathrm{and}\left(5,0.927+2\mathrm{k\pi }\right)$, for any integer k.