매개변수와 극좌표 전체 문제

Note that the distance from the point $\big(r\cos\theta\,,\,r\sin\theta\big)$ to the origin is

$$\sqrt{r^2\cos^2\theta + r^2\sin^2\theta} =\sqrt{r^2} =|r|$$

Thus $r$ can be either the distance to the origin or minus the distance to the origin.

(a) The distance from $(-2,0)$ to the origin is $2$. So either $r=2$ or $r=-2$.

• If $r=2$, then $\theta$ must obey

$$\begin{aligned} (-2,0) = \big(2\cos\theta\,,\,2\sin\theta\big) &\iff \sin\theta=0,\ \cos\theta=-1 \\ &\iff \theta= n\pi,\ n\text{ integer },\ \cos\theta=-1 \\ &\iff \theta= n\pi,\ n\text{ odd integer } \end{aligned}$$

• If $r=-2$, then $\theta$ must obey

$$\begin{aligned} (-2,0) = \big(-2\cos\theta\,,\,-2\sin\theta\big) &\iff \sin\theta=0,\ \cos\theta=1 \\ &\iff \theta= n\pi,\ n\text{ integer },\ \cos\theta=1 \\ &\iff \theta= n\pi,\ n\text{ even integer } \end{aligned}$$

In the figure on the left below, the blue half-line is the set of all points with polar coordinates $\theta=\pi$, $r>0$ and the pink half-line is the set of all points with polar coordinates $\theta=\pi$, $r<0$. In the figure on the right below, the blue half-line is the set of all points with polar coordinates $\theta=0$, $r>0$ and the pink half-line is the set of all points with polar coordinates $\theta=0$, $r<0$.

(b) The distance from $(1,1)$ to the origin is $\sqrt{2}$. So either $r=\sqrt{2}$ or $r=-\sqrt{2}$.

• If $r=\sqrt{2}$, then $\theta$ must obey

$$\begin{aligned} (1,1) = \big(\sqrt{2}\,\cos\theta\,,\,\sqrt{2}\,\sin\theta\big) &\iff \sin\theta=\cos\theta={1}/{\sqrt{2}} \\ &\iff \theta= {\pi}/{4} + 2n\pi,\ n\text{ integer } \end{aligned}$$

• If $r=-\sqrt{2}$, then $\theta$ must obey

$$\begin{aligned} (1,1) = \big(-\sqrt{2}\,\cos\theta\,,\,-\sqrt{2}\,\sin\theta\big) &\iff \sin\theta=\cos\theta=-{1}/{\sqrt{2}} \\ &\iff \theta= {5\pi}/{4} + 2n\pi,\ n\text{ integer } \end{aligned}$$

In the figure on the left below, the blue half-line is the set of all points with polar coordinates $\theta=\frac{\pi}{4}$, $r>0$ and the pink half-line is the set of all points with polar coordinates $\theta=\frac{\pi}{4}$, $r<0$. In the figure on the right below, the blue half-line is the set of all points with polar coordinates $\theta=\frac{5\pi}{4}$, $r>0$ and the pink half-line is the set of all points with polar coordinates $\theta=\frac{5\pi}{4}$, $r<0$.

(c) The distance from $(-1,-1)$ to the origin is $\sqrt{2}$. So either $r=\sqrt{2}$ or $r=-\sqrt{2}$.

• If $r=\sqrt{2}$, then $\theta$ must obey

$$\begin{aligned} (-1,-1) = \big(\sqrt{2}\,\cos\theta\,,\,\sqrt{2}\,\sin\theta\big) &\iff \sin\theta=\cos\theta=-{1}/{\sqrt{2}} \\ &\iff \theta= {5\pi}/{4} + 2n\pi,\ n\text{ integer } \end{aligned}$$

• If $r=-\sqrt{2}$, then $\theta$ must obey

$$\begin{aligned} (-1,-1) = \big(-\sqrt{2}\,\cos\theta\,,\,-\sqrt{2}\,\sin\theta\big) &\iff \sin\theta=\cos\theta={1}/{\sqrt{2}} \\ &\iff \theta= {\pi}/{4} + 2n\pi,\ n\text{ integer } \end{aligned}$$

In the figure on the left below, the blue half-line is the set of all points with polar coordinates $\theta=\frac{5\pi}{4}$, $r>0$ and the pink half-line is the set of all points with polar coordinates $\theta=\frac{5\pi}{4}$, $r<0$. In the figure on the right below, the blue half-line is the set of all points with polar coordinates $\theta=\frac{\pi}{4}$, $r>0$ and the pink half-line is the set of all points with polar coordinates $\theta=\frac{\pi}{4}$, $r<0$.