### Instruction: Let $A = (3, \theta_1)$ and $B = (9, \theta_2)$ in polar coordinates.  If $\theta_1 - \theta_2 = \frac{\pi}{2},$ then find the distance $AB.$.### Response: To find the distance between two points in polar coordinates, I need to use the polar distance formula, which is $AB = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1 - \theta_2)}.$
In this problem, I have $r_1 = 3, r_2 = 9, \theta_1 - \theta_2 = \frac{\pi}{2}.$
So I can plug these values into the formula and simplify.
$AB = \sqrt{3^2 + 9^2 - 2(3)(9)\cos(\frac{\pi}{2})}.$
I notice that $\cos(\frac{\pi}{2}) = 0,$ so the last term vanishes.
This makes the calculation easier.
$AB = \sqrt{9 + 81 - 0} = \sqrt{90}.$
I can simplify the square root by factoring out the largest perfect square factor, which is $9.$
$AB = \sqrt{9 \cdot 10} = \sqrt{9} \cdot \sqrt{10} = 3 \cdot \sqrt{10}.$