Find the distance between the points P left( | vedclass

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(A) The distance $d$ between two points $(r_1, 	heta_1)$ and $(r_2, 	heta_2)$ in polar coordinates is given by the formula: $d = \sqrt{r_1^2 + r_2^2

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$5$

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(A) The distance $d$ between two points $(r_1, 	heta_1)$ and $(r_2, 	heta_2)$ in polar coordinates is given by the formula: $d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2 \cos(	heta_1 - 	heta_2)}$ Given $r_1 = 3, 	heta_1 = -\frac{\pi}{6}$ and $r_2 = 4, 	heta_2 = \frac{\pi}{3}$. Substituting the values: $PQ = \sqrt{3^2 + 4^2 - 2(3)(4) \cos\left( -\frac{\pi}{6} - \frac{\pi}{3} ight)}$ $PQ = \sqrt{9 + 16 - 24 \cos\left( -\frac{\pi}{2} ight)}$ Since $\cos\left( -\frac{\pi}{2} ight) = 0$: $PQ = \sqrt{25 - 24(0)} = \sqrt{25} = 5$.

Find the distance between the points $P \left( 3, -\frac{\pi}{6} \right)$ and $Q \left( 4, \frac{\pi}{3} \right)$ in polar coordinates.

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