Question

How do you convert the rectangular coordinate `(-4.26,31.1)` into polar coordinates?

Answer 1

`(31.3,pi/2)`

Explanation:

Changing to polar coordinates means we have to find `color(green)((r,theta))`.

Knowing the relation between rectangular and polar coordinates that says:
`color(blue)(x=rcostheta and y=rsintheta)`

Given the rectangular coordinates:
`x=-4.26 and y=31.3`

`x^2+y^2=(-4.26)^2+(31.3)^2`
`color(blue)((rcostheta )^2)+color(blue)((rsintheta)^2)=979.69`
`r^2cos^2theta+r^2sin^2theta=979.69`
`r^2(cos^2theta+sin^2theta)=979.69`
Knowing the trigonometric identity that says:
`color(red)(cos^2theta+sin^2theta=1)`
We have:

`r^2*color(red)1=979.69`
`r=sqrt(979.69)`
`color(green)(r=31.3)`

Given:
`color(blue)y=31.3`
`color(blue)(rsintheta)=31.3`
`color(green)31.3*sintheta31.3`
`sintheta=31.3/31.3`
`sintheta=1`
`color(green)(theta=pi/2)`

Therefore,the polar coordinates are
`(color(green)(31.3,pi/2))`