Question

Convert the equations given in polar coordinates to equations in rectangular coordinates

$\theta =\frac{k\mathrm{\pi }}{6}$, for each positive integer k less than 12.

Short answer

The equations in the polar for for different value of k:

$$\begin{gathered} x=\sqrt{3}y,k=1 \\ y=\sqrt{3}x,k=2 \\ x=0,k=3 \\ y=-\sqrt{3}x,k=4 \\ x=-\sqrt{3}y,k=5 \\ y=0,k=6 \\ x=\sqrt{3}y,k=7 \\ y=\sqrt{3}x,k=8 \\ x=0,k=9 \\ y=-\sqrt{3}x,k=10 \\ x=-\sqrt{3}y,k=11 \end{gathered}$$

Step-by-step solution

Step 1. Given information

$\theta =\frac{k\mathrm{\pi }}{6}$,for each positive integer k less than 12.

Step 2. Write equation in rectangular form for k=1

As we know that in polar form

$\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

So, when $k=1$

role="math" localid="1652108770892" $$\begin{gathered} \theta =\frac{\mathrm{\pi }}{6} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{\mathrm{\pi }}{6} \\ \frac{y}{x}=\tan \left(\frac{\mathrm{\pi }}{6}\right) \\ \frac{y}{x}=\frac{1}{\sqrt{3}} \\ x=y\sqrt{3} \end{gathered}$$

Step 3. Write equation in the rectangular form for k=2

$$\begin{gathered} \theta =\frac{2\mathrm{\pi }}{6} \\ \theta =\frac{\mathrm{\pi }}{3} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{\mathrm{\pi }}{3} \\ \frac{y}{x}=\tan \left(\frac{\mathrm{\pi }}{3}\right) \\ \frac{y}{x}=\sqrt{3} \\ y=x\sqrt{3} \end{gathered}$$

Step 4. Write equation in the rectangular form for k=3

$$\begin{gathered} \theta =\frac{3\mathrm{\pi }}{6} \\ \theta =\frac{\mathrm{\pi }}{2} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{\mathrm{\pi }}{2} \\ \frac{y}{x}=\tan \left(\frac{\mathrm{\pi }}{2}\right) \\ \frac{y}{x}=\frac{1}{0} \\ x=0 \end{gathered}$$

Step 5. Write equation in the rectangular form for k=4

$$\begin{gathered} \theta =\frac{4\mathrm{\pi }}{6} \\ \theta =\frac{2\mathrm{\pi }}{3} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{2\mathrm{\pi }}{3} \\ \frac{y}{x}=\tan \left(\frac{2\mathrm{\pi }}{3}\right) \\ \frac{y}{x}=-\sqrt{3} \\ y=-x\sqrt{3} \end{gathered}$$

Step 6. Write equation in the rectangular form for k=5

$$\begin{gathered} \theta =\frac{5\mathrm{\pi }}{6} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{5\mathrm{\pi }}{6} \\ \frac{y}{x}=\tan \left(\frac{5\mathrm{\pi }}{6}\right) \\ \frac{y}{x}=-\frac{1}{\sqrt{3}} \\ x=-y\sqrt{3} \end{gathered}$$

Step 7. Write equation in the rectangular form for k=6

$$\begin{gathered} \theta =\frac{6\mathrm{\pi }}{6} \\ \theta =\mathrm{\pi } \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\mathrm{\pi } \\ \frac{y}{x}=\mathrm{tan\pi } \\ \frac{y}{x}=0 \\ y=0 \end{gathered}$$

Step 8. Write equation in the rectangular form for k=7

$$\begin{gathered} \theta =\frac{7\mathrm{\pi }}{6} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{7\mathrm{\pi }}{6} \\ \frac{y}{x}=\tan \left(\frac{7\mathrm{\pi }}{6}\right) \\ \frac{y}{x}=\frac{1}{\sqrt{3}} \\ x=y\sqrt{3} \end{gathered}$$

Step 9. Write equation in the rectangular form for k=8

$$\begin{gathered} \theta =\frac{8\mathrm{\pi }}{6} \\ \theta =\frac{4\mathrm{\pi }}{3} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{4\mathrm{\pi }}{3} \\ \frac{y}{x}=\tan \left(\frac{4\mathrm{\pi }}{3}\right) \\ \frac{y}{x}=\sqrt{3} \\ y=x\sqrt{3} \end{gathered}$$

Step 10. Write equation in the rectangular form for k=9

$$\begin{gathered} \theta =\frac{9\mathrm{\pi }}{6} \\ \theta =\frac{3\mathrm{\pi }}{2} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{3\mathrm{\pi }}{2} \\ \frac{y}{x}=\tan \left(\frac{3\mathrm{\pi }}{2}\right) \\ \frac{y}{x}=\frac{1}{0} \\ x=0 \end{gathered}$$

Step 11. Write equation in the rectangular form for k=10

$$\begin{gathered} \theta =\frac{10\mathrm{\pi }}{6} \\ \theta =\frac{5\mathrm{\pi }}{3} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{5\mathrm{\pi }}{3} \\ \frac{y}{x}=\tan \left(\frac{5\mathrm{\pi }}{3}\right) \\ \frac{y}{x}=-\sqrt{3} \\ y=-x\sqrt{3} \end{gathered}$$

Step 12. Write equation in the rectangular form for k=11

$$\begin{gathered} \theta =\frac{11\mathrm{\pi }}{6} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{11\mathrm{\pi }}{6} \\ \frac{y}{x}=\tan \left(\frac{11\mathrm{\pi }}{6}\right) \\ \frac{y}{x}=-\frac{1}{\sqrt{3}} \\ x=-y\sqrt{3} \end{gathered}$$