Chapter 7, Problem 322RE

True or False? Justify your answer with a proof or a counterexample.

322. The rectangular coordinates of the point $\left(4,\frac{5\pi }{6}\right)$ are $\left(2\sqrt{3},-2\right)$ .

To determine

To check: The given statement whether it is true or false and also justified it with a proof.

Answer to Problem 322RE

The given statement is false because the rectangular coordinates of $\left(4,\frac{5\pi }{6}\right)$ are $\left(-2\sqrt{3},2\right)$ .

Explanation of Solution

Given information:

The polar coordinates are $\left(4,\frac{5\pi }{6}\right)$ and the rectangular coordinates are $\left(2\sqrt{3},-2\right)$ .

Concept used:

The polar coordinates of any rectangular coordinates $\left(x,y\right)$ are $\left(r,\theta \right)$ , where $x=r\cos \theta \text{ and }y=r\sin \theta$ .

Calculation:

The given polar coordinates are $\left(4,\frac{5\pi }{6}\right)$ .

$r=4$ ........ $\left(1\right)$

$\theta =\frac{5\pi }{6}$ ........ $\left(2\right)$

Substitute 4 for $r$ and $\frac{5\pi }{6}$ for $\theta$ in the formula $x=r\cos \theta .$

  $\begin{array}{c}x=4\cos \frac{5\pi }{6}\\ =4\cos \left(\pi -\frac{\pi }{6}\right)\\ =-4\cos \frac{\pi }{6}\left[\text{Using }\cos \left(\pi -\theta \right)\text{=}-\cos \theta \right]\\ =-4\left(\frac{\sqrt{3}}{2}\right)\left[\text{Using }\cos \frac{\pi }{6}\text{=}\frac{\sqrt{3}}{2}\right]\\ =-2\sqrt{3}\end{array}$

Substitute 4 for $r$ and $\frac{5\pi }{6}$ for $\theta$ in the formula $x=r\cos \theta .$

$\begin{array}{c}y=4\sin \frac{5\pi }{6}\\ =4\sin \left(\pi -\frac{\pi }{6}\right)\\ =4\sin \frac{\pi }{6}\left[\text{Using sin}\left(\pi -\theta \right)=\sin \theta \right]\\ =4\left(\frac{1}{2}\right)\left[\text{Using }\sin \frac{\pi }{6}\text{=}\frac{1}{2}\right]\\ =2\end{array}$

The required rectangular coordinates are $\left(-2\sqrt{3},2\right)$ .

Conclusion:

Hence, the given statement is false.