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

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

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Answer 1 · 2016-11-24T11:10:54

$(\sqrt{65}, - 1.05)$ $(sqrt65,-1.05)$

Explanation:

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

That is $(x, y) \to (r, θ)$ $(x,y)to(r,theta)$

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

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 $theta=-1.05larr" in 4th quadrant"$

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

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

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