Convert the Cartesian coordinate $(-3,6)$ to polar coordinates, $0 \leq \theta< 2 \pi$
\[
r=
\]
Enter exact value.
\[
\theta=
\]

Answer

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Answer

$\boxed{r = \sqrt{45}, \theta = \arctan(-2) + \pi}$

Steps

Step 1 ：Find the distance from the origin (r) using the formula r = $\sqrt{x^2 + y^2}$: $r = \sqrt{(-3)^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45}$

Step 2 ：Find the angle (θ) using the formula θ = arctan(y/x): $\theta = \arctan\left(\frac{6}{-3}\right) = \arctan(-2)$

Step 3 ：Since the point is in the second quadrant, add π to the angle: $\theta = \arctan(-2) + \pi$

Step 4 ：The polar coordinates for the Cartesian coordinate (-3, 6) are: $r = \sqrt{45}$ and $\theta = \arctan(-2) + \pi$

Step 5 ：$\boxed{r = \sqrt{45}, \theta = \arctan(-2) + \pi}$