How do you convert the Cartesian coordinates (0,-10) to polar coordinates?

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Answer 1 · 2015-09-12T22:12:07

$(r, θ) = (10, \frac{3 π}{2})$ $(r,theta)=(10,(3pi)/2)$

When converting from Cartesian to polar coordinates we use the following relationships

$x = r cos θ$ $x=rcostheta$

$y = r sin θ$ $y=rsintheta$

$r = \sqrt{x^{2} + y^{2}}$ $r=sqrt(x^2+y^2)$

We are given the point $(0, - 10)$ $(0,-10)$

Proceed by finding $r$ $r$ . Since $r$ $r$ is a length we only want the positive square root.

$r = \sqrt{{(0)}^{2} + {(- 10)}^{2}}$ $r=sqrt((0)^2+(-10)^2)$

$r = \sqrt{100} = 10$ $r=sqrt100=10$

Now we utilize the other relationships

$x = r cos θ$ $x=rcostheta$ so we can write

$0 = 10 cos θ$ $0=10costheta$

So $cos θ = 0$ $costheta=0$

We know that we are on the y axis at $x = 0$ $x=0$ and $y = - 10$ $y=-10$ so our angle is

$θ = \frac{3 π}{2}$ $theta=(3pi)/2$

Also

$y = r sin θ$ $y=rsintheta$ so we can write

$- 10 = 10 sin θ$ $-10=10sintheta$

$sin θ = - 1$ $sintheta=-1$

This occurs when $θ = \frac{3 π}{2}$ $theta=(3pi)/2$

Therefore $(r, θ) = (10, \frac{3 π}{2})$ $(r,theta)=(10,(3pi)/2)$

Make sure your calculator is in radians