QuestionUse the Distance Formula $d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)}$ to find the distance between the two points in polar coordinates. (Round your answer to one decimal place.) $$\left(2, \frac{5 \pi}{6}\right),\left(5, \frac{\pi}{3}\right)$$

Studdy Solution
Calculate the difference in angles:
$$\theta_{1} - \theta_{2} = \frac{5\pi}{6} - \frac{\pi}{3} = \frac{5\pi}{6} - \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}$$
Substitute back into the formula:
$$d = \sqrt{2^{2} + 5^{2} - 2 \times 2 \times 5 \times \cos\left(\frac{\pi}{2}\right)}$$
Since $\cos\left(\frac{\pi}{2}\right) = 0$, the formula simplifies to:
$$d = \sqrt{2^{2} + 5^{2}}$$ $$= \sqrt{4 + 25}$$ $$= \sqrt{29}$$
Calculate and round to one decimal place:
$$d \approx 5.4$$
The distance between the two points is:
$$\boxed{5.4}$$