BAB 5 : PERSAMAAN PARAMETRIK DAN KOODINAT KUTUB

A) Rangkuman Materi

1 Persamaan Karakteristik untuk Kurva Kutub

\(x = rcos \theta y=rsin \theta\)

2 Garis Singgung pada Kurva Kutub

Jika \(r = f(\theta)\), maka

\[ \begin{aligned} \frac{dx}{d\theta} &= -r \sin\theta + \frac{dr}{d\theta} \cos\theta \\ \frac{dy}{d\theta} &= r \cos\theta + \sin\theta \frac{dr}{d\theta} \\ \frac{dy}{dx} &= \frac{r \cos\theta + \sin\theta \frac{dr}{d\theta}}{-r \sin\theta + \cos\theta \frac{dr}{d\theta}} \quad \text{(1)} \end{aligned} \]

Kemiringan garis singgung kurva \(r = f(\theta)\) di titik \((r, \theta)\) ditunjukkan oleh persamaan (1).

3 Panjang Busur Kurva Kutub

B) Contoh Soal

1. Dapatkan kemiringan garis singgung dari kurva \(r = \frac{1}{\theta}\) di titik dengan \(\theta = 2\).

Pembahasan:

\[ \begin{aligned} \frac{dy}{dx} &= \frac{\frac{1}{\theta} \cos\theta + \sin\theta \frac{d}{d\theta}\left[\frac{1}{\theta}\right]}{-\frac{1}{\theta} \sin\theta + \cos\theta \frac{d}{d\theta}\left[\frac{1}{\theta}\right]}\Bigg|_{\theta=2} \\ &= \frac{\frac{1}{\theta} \cos\theta + \sin\theta \left[-\frac{1}{\theta^2}\right]}{-\frac{1}{\theta} \sin\theta + \cos\theta \left[-\frac{1}{\theta^2}\right]}\Bigg|_{\theta=2} \\ &= \frac{\theta \cos\theta - \sin\theta}{-\theta \sin\theta - \cos\theta}\Bigg|_{\theta=2} \\ &= \frac{2 \cos 2 - \sin 2}{-2 \sin 2 - \cos 2} \end{aligned} \]

2. Soal EAS 2020

Dapatkan kemiringan garis singgung dari kurva \(r = 3\sin(3\theta)\) di titik dengan \(\theta = \frac{\pi}{6}\).

Pembahasan:

\[ \begin{aligned} \frac{dy}{dx} &= \frac{3\sin(3\theta) \cos\theta + \sin\theta \frac{d}{d\theta}[3\sin(3\theta)]}{-3\sin(3\theta) \sin\theta + \cos\theta \frac{d}{d\theta}[3\sin(3\theta)]}\Bigg|_{\theta=\frac{\pi}{6}} \\ &= \frac{3\sin(3\theta) \cos\theta + \sin\theta\, 9\cos(3\theta)}{-3\sin(3\theta) \sin\theta + \cos\theta\, 9\cos(3\theta)}\Bigg|_{\theta=\frac{\pi}{6}} \\ &= \frac{\sin(3\theta) \cos\theta + 3\sin\theta\cos(3\theta)}{-\sin(3\theta) \sin\theta + 3\cos\theta\cos(3\theta)}\Bigg|_{\theta=\frac{\pi}{6}} \\ &= \frac{\sin(\frac{\pi}{2}) \cos\frac{\pi}{6} + 3\sin\frac{\pi}{6}\cos\frac{\pi}{2}}{-\sin(\frac{\pi}{2}) \sin\frac{\pi}{6} + 3\cos\frac{\pi}{6}\cos\frac{\pi}{2}} \\ &= \frac{1 \cdot \frac{\sqrt{3}}{2} + 3 \cdot \frac{1}{2} \cdot 0}{-1 \cdot \frac{1}{2} + 3 \cdot \frac{\sqrt{3}}{2} \cdot 0} \\ &= \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} \\ &= -\sqrt{3} \end{aligned} \]

Jadi, kemiringan garis singgung dari kurva \(r = 3\sin(3\theta)\) di titik dengan \(\theta = \frac{\pi}{6}\) adalah \(-\sqrt{3}\).

3. Dapatkan panjang busur dari kurva \(r = e^{3\theta}\) dari \(\theta = 0\) sampai \(\theta = 2\).

Pembahasan:

\[ \begin{aligned} S &= \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta \\ &= \int_{0}^{2} \sqrt{(e^{3\theta})^2 + \left(\frac{d}{d\theta}[e^{3\theta}]\right)^2} d\theta \\ &= \int_{0}^{2} \sqrt{e^{6\theta} + (3e^{3\theta})^2} d\theta \\ &= \int_{0}^{2} \sqrt{e^{6\theta} + 9e^{6\theta}} d\theta \\ &= \int_{0}^{2} \sqrt{10e^{6\theta}} d\theta \\ &= \sqrt{10} \int_{0}^{2} e^{3\theta} d\theta \\ &= \sqrt{10} \left( \frac{1}{3}e^{3\theta} \right)\Bigg|_{0}^{2} \\ &= \frac{\sqrt{10}}{3} (e^6 - 1) \end{aligned} \]

C) Latihan Soal