Chapter H.1, Problem 4E

(a)

To determine

To plot: the point and find the Cartesian coordinates of the point.

Answer to Problem 4E

Cartesian coordinates of $\left(-\sqrt{2},\frac{5\pi }{4}\right)=(1,1)$

Explanation of Solution

Given:

Polar coordinates are : $(-\sqrt{2},5\pi /4)$

Calculation:

  

  $\left(-\sqrt{2},\frac{5\pi }{4}\right)=(r,\theta )\Rightarrow r=-\sqrt{2};\theta =\frac{5\pi }{4}$

  $x=r\cos \theta =\sqrt{2}\cdot \cos \left(\frac{5\pi }{4}\right)$

  $=-\sqrt{2}\cdot \cos \left(\pi +\frac{\pi }{4}\right)$

  $=-\sqrt{2}\cdot \left(-\cos \frac{\pi }{4}\right)$

  $=-\sqrt{2}\cdot \left(\frac{-1}{\sqrt{2}}\right)$

  $=1$

  $y=r\sin \theta =-\sqrt{2}\cdot \sin \left(\frac{5\pi }{4}\right)$

  $=-\sqrt{2}\cdot \sin \left(\pi +\frac{\pi }{4}\right)$

  $=-\sqrt{2}\cdot \left(-\sin \frac{\pi }{4}\right)$

  $=-\sqrt{2}\cdot \left(\frac{-1}{\sqrt{2}}\right)$

  $=1$

  $\therefore$ Cartesian coordinates of $\left(-\sqrt{2},\frac{5\pi }{4}\right)=(1,1)$

Conclusion:

Therefore, the Cartesian coordinates of $\left(-\sqrt{2},\frac{5\pi }{4}\right)=(1,1)$

(b)

To determine

To plot: the point and find the Cartesian coordinates of the point.

Answer to Problem 4E

Cartesian coordinates of $\left(1,\frac{5\pi }{2}\right)=(0,1)$

Explanation of Solution

Given:

Polar coordinates are: $(1,5\pi /2)$

Calculation:

  

  $(r,\theta )=\left(1,\frac{5\pi }{2}\right)\Rightarrow r=1,\theta =\frac{5\pi }{2}$

  $\therefore x=r\cos \theta =1\cdot \cos \left(\frac{5\pi }{2}\right)$

  $=\cos \left(2\pi +\frac{\pi }{2}\right)$

  $=\cos \frac{\pi }{2}$

  $=0$

  $y=r\sin \theta$

  $=1\cdot \sin \left(\frac{5\pi }{2}\right)$

  $=\sin \left(2\pi +\frac{\pi }{2}\right)$

  $=\sin \frac{\pi }{2}$

  $=1$

  $\therefore$ Cartesian coordinates of $\left(1,\frac{5\pi }{2}\right)=(0,1)$

Conclusion:

Therefore, the Cartesian coordinates of $\left(1,\frac{5\pi }{2}\right)=(0,1)$

(c)

To determine

To plot: the point and find the Cartesian coordinates of the point.

Answer to Problem 4E

The Cartesian coordinates are $(x,y)=(-\sqrt{3},1)$

Explanation of Solution

Given:

Polar coordinates are: $(2,-7\pi /6)$

Calculation:

  

  $\left(2,\frac{-7\pi }{6}\right)=(r,\theta )\Rightarrow r=2;0=\frac{-7\pi }{6}$

  $x=r\cos \theta =2\cdot \cos \left(\frac{-7\pi }{6}\right)$

  $=2\cos \left(\frac{7\pi }{6}\right)$

  $=2\cos \left(\pi +\frac{\pi }{6}\right)$

  $=2\left[-\cos \frac{\pi }{6}\right](\because \cos (\pi +\theta )=-\cos \theta )$

  $=-2\times \frac{\sqrt{3}}{2}$

  $=-\sqrt{3}$

  $y=r\sin \theta =2\cdot \sin \left(\frac{-7\pi }{6}\right)$

  $=-2\sin \left(\frac{7\pi }{6}\right)$

  $=-2\cdot \sin \left(\pi +\frac{\pi }{6}\right)$

  $=-2\cdot \left[-\sin \frac{\pi }{6}\right](\because \sin (\pi +\theta )=-\sin \theta )$

  $=2\sin \frac{\pi }{6}$

  $=2\cdot \frac{1}{2}$

  $=1$

  $\therefore$ The Cartesian coordinates of $\left(2,\frac{-7\pi }{6}\right)$

  $=(x,y)=(-\sqrt{3},1)$

Conclusion:

Therefore, the Cartesian coordinates are $(x,y)=(-\sqrt{3},1)$