Question

Rectangular Coordinates to Polar Coordinates

Convert the rectangular coordinates to polar coordinates with$r>0$and

$0\le \theta <2\pi$$(1,-2)$

Short answer

The polar coordinates are$(r,\theta )=(\sqrt{5},1.648\pi )$

Step-by-step solution

Step 1. Given information

A point is given with rectangular coordinates.$(1,-2)$$(1,-2)$

Step 2. Concept used

Replace the values in the equations and do the computations. There may be many polar points that emerge as responses. However, the right response is one that falls inside the question's range.

Step 3. Calculation

The point given is$(x,y)=(1,-2)$, we use formulae as-

$\begin{array}{l}{r}^2=x^2+y^2\\ \theta ={\tan }^{-1}\left(\frac{y}{x}\right)\end{array}$

Now substitute the values in formulae,

$\begin{array}{l}\theta ={\tan }^{-1}\left(\frac{y}{x}\right)\\ \theta ={\tan }^{-1}\left(\frac{-2}{1}\right)\\ \theta ={\tan }^{-1}(-2)\\ \theta =0.648\pi ,1.648\pi \end{array}$

Since,$r>0$we consider$r=\sqrt{5}$only and this occurs at.$\theta =1.648\pi$$\theta =1.648\pi$