The area (in square unit) of the triangle formed by the points with polar coordinates $(1,0),\left(2,…
The area (in square unit) of the triangle formed by the points with polar coordinates $(1,0),\left(2, \frac{\pi}{3}\right)$ and $\left(3, \frac{2 \pi}{3}\right)$ is
$\frac{11 \sqrt{3}}{4}$
$\frac{5 \sqrt{3}}{4}$
$\frac{5}{4}$
$\frac{11}{4}$
Solution
Area of the triangle formed by the points with polar coordinates $(1,0),\left(2, \frac{\pi}{3}\right)$ and $\left(3, \frac{2 \pi}{3}\right)$ is
$\begin{aligned} & \frac{1}{2}\left|\Sigma r_1 r_2 \sin \left(\theta_1-\theta_2\right)\right| \\ & =\frac{1}{2} \mid 2 \cdot 1 \sin \left(0-\frac{\pi}{3}\right)+2 \cdot 3 \sin \left(\frac{\pi}{3}-\frac{2 \pi}{3}\right) \\ & +3 \cdot 1 \sin \left(\frac{2 \pi}{3}-0\right) \\ & =\frac{1}{2}\left|-2 \cdot \frac{\sqrt{3}}{2}-\frac{6 \sqrt{3}}{2}+\frac{3 \sqrt{3}}{2}\right| \\ & =\frac{1}{2}\left|\frac{-5 \sqrt{3}}{2}\right| \\ & =\frac{5 \sqrt{3}}{4} \text { sq unit } \\ & \end{aligned}$