two cell phone towers were used to determine the location of an individual calling for assistance. the…

two cell phone towers were used to determine the location of an individual calling for assistance. the signal from one tower is represented by $r = -\frac{sqrt{3}}{2}+sin(\theta)$, and the signal from the other tower by $r = 2sin(\theta)$. the points of intersection of the curves are the possible locations of the individual. which coordinate is one of the possible locations?\n$(-sqrt{3},\frac{7pi}{6})$\n$(-sqrt{3},\frac{5pi}{3})$\n$(sqrt{3},\frac{7pi}{6})$\n$(sqrt{3},\frac{5pi}{3})

two cell phone towers were used to determine the location of an individual calling for assistance. the signal from one tower is represented by $r = -\frac{sqrt{3}}{2}+sin(\theta)$, and the signal from the other tower by $r = 2sin(\theta)$. the points of intersection of the curves are the possible locations of the individual. which coordinate is one of the possible locations?\n$(-sqrt{3},\frac{7pi}{6})$\n$(-sqrt{3},\frac{5pi}{3})$\n$(sqrt{3},\frac{7pi}{6})$\n$(sqrt{3},\frac{5pi}{3})

Answer

Answer:

A. $\left(-\sqrt{3},\frac{7\pi}{6}\right)$

Explanation:

Step1: Set the two equations equal

$-\frac{\sqrt{3}}{2}+\sin(\theta)=2\sin(\theta)$

Step2: Solve for $\sin(\theta)$

$-\frac{\sqrt{3}}{2}=2\sin(\theta)-\sin(\theta)$ $-\frac{\sqrt{3}}{2}=\sin(\theta)$

Step3: Find $\theta$ values

We know that $\sin(\theta)=-\frac{\sqrt{3}}{2}$ has solutions $\theta=\frac{4\pi}{3}+2k\pi$ and $\theta=\frac{5\pi}{3}+2k\pi,k\in\mathbb{Z}$. Among the given options, when $\theta = \frac{7\pi}{6}$, $\sin(\theta)=-\frac{1}{2}$. Substitute $\theta=\frac{7\pi}{6}$ into $r = 2\sin(\theta)$: $r=2\times(-\frac{1}{2})=-\sqrt{3}$ (note: in polar - coordinates, $r$ can be negative). So the point $\left(-\sqrt{3},\frac{7\pi}{6}\right)$ is a possible solution.