Question

How do you find the polar coordinates given the rectangular coordinates (4,-7)?

Answer 1

`(sqrt65,-1.05)`

Explanation:

To convert from `color(blue)"cartesian to polar coordinates"`

That is `(x,y)to(r,theta)`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(r=sqrt(x^2+y^2))color(white)(2/2)|)))`

and `color(red)(bar(ul(|color(white)(2/2)color(black)(theta=tan^-1(y/x))color(white)(2/2)|)))`
where `-pi < theta<=pi`

Here x = 4 and y = - 7

`rArrr=sqrt(4^2+(-7)^2)=sqrt(16+49)=sqrt65`

Now, (4 ,-7) is in the 4th quadrant so we must ensure that `theta` is in the 4th quadrant.
To calculate the `color(blue)"reference angle"` we can ignore the negative.

`rArrtheta=tan^-1(7/4)=1.05larr" reference angle"`

Thus `theta=-1.05larr" in 4th quadrant"`

`rArr(4,-7)to(sqrt65,-1.05)`