Chapter 9.5, Problem 41E

To determine

To calculate: To find one set of polar coordinatesfor the point in rectangular coordinates

Answer to Problem 41E

The polar coordinates for a point in rectangular coordinate is $\left(3.61,-0.588\right)$

Explanation of Solution

Given information: Rectangularis $\left(3,-2\right)$

Formula Used:

Polar coordinate of a point $\left(x,y\right)$ is given as

  $\left(r,\theta \right)$

Where,

  $r=\sqrt{x^2+y^2}$

  $\theta =\arctan \left(\left|\frac{y}{x}\right|\right)$

If both $x,y>0$ , then $\theta =\alpha$

If $x<0,y>0$ , then $\theta =\pi -\alpha$

If $x,y<0$ , then $\theta =\pi +\alpha$

If $x>0,y<0$ , then $\theta =-\alpha$

Calculation:

Rectangular point is given as

  $\left(3,-2\right)$

Converting rectangular coordinates into polar coordinates:

Polar coordinate of a point $\left(x,y\right)$ is given as

  $\left(r,\theta \right)$

Calculating the value of $r$ :

  $\begin{array}{l}r=\sqrt{{\left(3\right)}^2+{\left(-2\right)}^2}\\ r=\sqrt{9+4}\\ r=\sqrt{13}\\ r=3.61\end{array}$

Calculating the value of $\theta$ :

Since $x>0,y<0$ , then $\theta =-\alpha$

  $\begin{array}{l}\theta =-\arctan \left(\left|\frac{-2}{3}\right|\right)\\ \theta \approx -0.588\end{array}$

Thus, polar coordinate of point is $\left(3.61,-0.588\right)$

Conclusion:

Hence, polar coordinates for a point in rectangular coordinate is $\left(3.61,-0.588\right)$