Question

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

Answer 1

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*2} \\ r=5\sqrt[]{2} \end{gathered}`

The angle theta becomes

`\begin{gathered} \tan \theta=(-5)/(5) \\ \tan \theta=-1 \\ \theta=\tan ^(-1)(-1) \\ \theta=-45\degree \\ 180\degree=\pi\text{ radians } \\ -45\degree=(-45)/(180)\pi\text{ radians} \\ =-(1)/(4)\pi \\ =-(\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-(\pi)/(4) \\ =(7\pi)/(4) \end{gathered}`

Hence the answer becomes

`(5\sqrt[]{2},(7\pi)/(4))`

The second option is the correct answer