극좌표와 극방정식의 그래프

(a) Since $-1\le \sin(4\theta)\le 1$, the coordinate $r=2+\sin(4\theta)$ oscillates between $r=1$ and $r=3$ as $\theta$ runs from $0$ to $2\pi$. The maximum value $r=3$ is achieved when $\sin(4\theta)=1$, i.e when $4\theta=\frac{\pi}{2} +2n\pi$, i.e. when $\theta=\frac{\pi}{8}+\frac{n\pi}{2}$. That matches figure (E).

(b) Since $-1\le \sin(4\theta)\le 1$, the coordinate $r=1+2\sin(4\theta)$ takes its maximum value $r=3$ when $\sin(4\theta)=1$, i.e. when $\theta=\frac{\pi}{8}+\frac{n\pi}{2}$, just as the case with $(a)$. But now $r$ can also take the value $0$. That matches figure (B).

(c) $r=1$ is completely indepedent of $\theta$. All points on the curve $r=1$ are a distance $1$ from the origin. That is, $r=1$ is the circle of radius $1$ centred on the origin. That's figure (F).

(d) In this case, $\theta$ is subject to the restriction $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$, like figure (C). Figure (C) looks like a circle. We can verify that $r=2\cos(\theta)$ is indeed a circle by converting to Cartesian coordinates. We can convert the right hand side to exactly $2x=2r\cos(\theta)$ by multiplying the whole equation by $r$.

$$\begin{aligned} r=2\cos(\theta) &\iff r^2 =2r\cos(\theta) \iff x^2+y^2=2x \\ &\iff (x-1)^2+y^2=1 \end{aligned}$$

So $r=2\cos(\theta)$ is the circle of radius $1$ centred on $x=1$, $y=0$, which indeed matches figure (C).

(e) When $\theta=0$, $r=e^{\theta/10}+e^{-\theta/10}=2$. As

$$\frac{\mathrm{d}}{\mathrm{d}\theta}\big(e^{\theta/10}+e^{-\theta/10}\big) =\frac{1}{10}\big(e^{\theta/10}-e^{-\theta/10}\big) >0\qquad\text{for all }\theta > 0$$

$r=e^{\theta/10}+e^{-\theta/10}$ increases as $\theta$ increases for all $\theta\ge 0$. Furthermore the rate of increase gets bigger and bigger as $\theta$ gets bigger and bigger. So $r$ starts at $r=2$ when $\theta=0$ and increases faster and faster as $\theta$ increases. That matches figure (A).

(f) When $\theta=0$, $r=\theta=0$. As

$$\frac{\mathrm{d}}{\mathrm{d}\theta} \theta =1 \qquad\text{for all }\theta$$

$r=\theta$ increases as $\theta$ increases for all $\theta\ge 0$. Furthermore the rate of increase is independent of $\theta$. So $r$ starts at $r=0$ when $\theta=0$ and increases at a constant rate as $\theta$ increases. That matches figure (D).