Q. 15 - Calculus

Question

Find all polar coordinates that represent the point $(-1,0)$ that is also given in polar coordinates.

Step-by-Step Solution

Answer

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

Step 1: Given information

The rectangular coordinate $(-1,0)$.

Step 2: Calculation

Consider the rectangular coordinate $(-1,0)$.

The objective is to convert the rectangular coordinates to polar coordinates.

In the coordinate $(-1,0), x=-1$ and $y=0$.

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

Then

$$\begin{gathered} r=\sqrt{(-1{)}^2+(0{)}^2}[\text{ since }x=-1,y=0] \\ r=\sqrt{1+0} \\ r=\sqrt{1} \\ r=\pm 1 \end{gathered}$$

To calculate $\theta$ use the formula $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

By substituting $x, y$ in the formula,

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{0}{-1}\right)[sincex=-1,y=0] \\ \theta ={\tan }^{-1}\left(0\right) \\ \theta =\pi ,2\pi \end{gathered}$$

Take $r=-1,\theta =\pi$.

Then $(r,\theta )=(-1,\pi )$

Thus, the coordinates of $(r,\theta )$ are $(-1,2k\pi )$, for any integer $k$.

Take $r=1,\theta =2\pi$

Then $(r,\theta )=(1,2\pi )$.

Thus, the coordinates of $(r,\theta )$ are $(1,(2k+1\left)\pi \right)$, for any integer $k$.

Therefore, the polar coordinates are $(1,(2k+1\left)\pi \right)(-1,2k\pi )$, for any integer $k$.