Question

Which polar coordinates represent the same point as the rectangular coordinate (2,-1?

Answer 1

`\bf (\stackrel{a}{2}~,~\stackrel{b}{-1})\qquad \begin{cases} r=√(a^2+b^2)\\\\ \theta =tan^(-1)\left( (b)/(a) \right) \end{cases} \\\\[-0.35em] ~\dotfill\\\\ r=√(2^2+(-1)^2)\implies r=√(5) \\\\\\ \theta =tan^(-1)\left( \cfrac{-1}{2} \right)\implies \theta \approx -26.57^o\implies \theta \approx 333.43^o \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill (√(5)~~,~~333.43^o)~\hfill`

Answer 2

`(r,\theta); (√(5) , tan^(-1)((x)/(y)))\\(r,\theta); (-√(5) , -tan^(-1)((x)/(y)))`

Explanation:

Here we are given our rectangular coordinates as (2,-1) . We have to convert this into polar coordinates. The formula for conversion into polar form is

`r=√(x^2+y^2)`

`\theta=tan^(-1)((x)/(y))`

Substituting the values of x and y in the above formulas we get

`r=√(2^2+(-1)^2)\\r=√(4+1)\\r=√(5)\\r=-√(5)\\`

`\theta=tan^(-1)((-1)/(2))`

Hence our polar coordinates are

`r=(√(5),tan^(-1)((-1)/(2)) )\\r=(-√(5),tan^(-1)((-1)/(2)) )\\`