Question

Find a pair of polar coordinates for the point with rectangular coordinates (5, –5).

Answer 1

SOLUTION

We want to find a pair of polar coordinates for the point with rectangular coordinates (5, –5).

This simply means we should find

$\begin{gathered} (r,\alpha) \\ \text{Where r is the magnitude of the points (5, -5) } \\ \alpha\text{ is the angle of r} \end{gathered}$

Let's represent this using the diagram below

From the diagram above, we can obtain r using Pythagoras theorem, this becomes

$\begin{gathered} r^2=(-5)^2+5^2 \\ r=\sqrt[]{25+25} \\ r=\sqrt[]{50} \\ r=\sqrt[]{25\times2} \\ r=5\sqrt[]{2} \end{gathered}$

The angle theta becomes

$\begin{gathered} \tan \theta=\frac{-5}{5} \\ \tan \theta=-1 \\ \theta=\tan ^{-1}(-1) \\ \theta=-45\degree \\ 180\degree=\pi\text{ radians } \\ -45\degree=\frac{-45}{180}\pi\text{ radians} \\ =-\frac{1}{4}\pi \\ =-\frac{\pi}{4}\text{ radians } \end{gathered}$

from our options, the alpha should be the required angle, we have

$\begin{gathered} 360\degree=2\pi\text{ radians } \\ 2\pi-\frac{\pi}{4} \\ =\frac{7\pi}{4} \end{gathered}$

Hence the answer becomes

$(5\sqrt[]{2},\frac{7\pi}{4})$

The second option is the correct answer