Q. 13 - Calculus

Question

Find all polar coordinates that represent the point $(1,0)$ given in rectangular 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: Simplification

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,

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

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

By substituting $x, y$ in the formula,

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

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

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

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

Now take $r=1,\theta =2\pi$

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

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

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