Chapter 8.1, Problem 26E

To determine

To calculate: To find the rectangular coordinates for the polar form

Answer to Problem 26E

The rectangular coordinates are $\left(-\sqrt{3},1\right)$ .

Explanation of Solution

Given: Polar coordinates of point is

  

Formula Used:

A polar equation is any equation that describes a relation between r and θ, where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Also, $x=r\cos \theta$ and $y=r\sin \theta$

  $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Calculation:

Polar coordinates of point is given as follows:

  

Here, polar coordinates are $R\left(2,\frac{5\pi }{6}\right)$

Using the above definition, we have:

  $r=2,\theta =\frac{5\pi }{6}$

  $\begin{array}{l}x=r\cos \theta \\ x=2\cos \left(\frac{5\pi }{6}\right)\\ x=-2\times \frac{\sqrt{3}}{2}\\ x=-\sqrt{3}\end{array}$

Now,

  $\begin{array}{l}y=r\sin \theta \\ y=2\times \sin \left(\frac{5\pi }{6}\right)\\ y=2\times \frac{1}{2}\\ y=1\end{array}$

Thus, the rectangular coordinates are $\left(-\sqrt{3},1\right)$ .

Conclusion:

The rectangular coordinates are $\left(-\sqrt{3},1\right)$ .