Q. 14 - Calculus

Question

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

Step-by-Step Solution

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 1: Given information

 The rectangular coordinate $(0,1)$

Step 2: 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 $k$.