Question

Rectangular Coordinates to Polar Coordinates

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

$0\le \theta <2\pi$$(3,4)$

Short answer

The polar coordinates are$(r,\theta )=(5,0.295\pi )$

Step-by-step solution

Step 1. Given information

A point is given with rectangular coordinates.$(3,4)$$(3,4)$

Step 2. Concept used

Substitute the values in the formulae, do calculations. There may be more than one polar points come out as answers. But the correct answer is that which is within the range given in question.

Step 3. Calculation

The point given is$(x,y)=(3,4)$, 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,

$\begin{array}{l}{r}^2=x^2+y^2\\ \Rightarrow r^2=3^2+4^2\\ \Rightarrow r^2=25\\ \Rightarrow r=\pm 5\end{array}$

$\begin{array}{l}\theta ={\tan }^{-1}\left(\frac{y}{x}\right)\\ \theta ={\tan }^{-1}\left(\frac{4}{3}\right)\\ \theta =0.295\pi ,1.295\pi \end{array}$

Since $r>0$, we consider $r=5$only and this occurs at $\theta =0.295\pi$.