Question

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

Short answer

The polar coordinates that represent the point $(0,1)$are $(-1,-\frac{\pi }{2}+2k\pi )$and $(1,\frac{\pi }{2}+2k\pi )$, for any integer K.

Step-by-step solution

Given information

The rectangular coordinate $(0,1)$

Calculation

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

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

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

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

Then,$r=\sqrt{x^2+y^2}$

$r=\sqrt{x^2+y^2}$

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

$r=\pm 1$

By substituting $x,y$in the formula,

$\theta ={\tan }^{-1}\left(\frac{1}{0}\right)$[since $x=0,y=1]$

$\theta ={\tan }^{-1}(\infty )$

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

Take $r=-1,\theta =-\frac{\pi }{2}$.

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

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

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

$(r,\theta )=\left(1,\frac{\pi }{2}\right)$

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