91Ó°ÊÓ

First, start by finding the derivative of the given polar equation with respect to \(\theta\). The given equation is \(r = \sin(2 \theta)\). To find \(dr/d\theta\), differentiate the polar equation with respect to \(\theta\): $$\frac{dr}{d\theta} = \frac{d}{d\theta}(\sin 2\theta) = 2\cos 2\theta$$

To find the points where we have a horizontal tangent, we must find the change of the angle (\(\frac{d\theta}{dr}\)) with respect to the radius. From the given equation, \(r = \sin(2\theta)\), we can express the angle function as \(\theta = \frac{1}{2}\arcsin r\). Now differentiate the angle function with respect to the radius, \(r\): $$\frac{d\theta}{dr} = \frac{1}{2}\frac{d}{dr}(\arcsin r) = \frac{1}{2}\frac{1}{\sqrt{1-r^2}}$$ Set this expression equal to zero to find points where we have a horizontal tangent. Notice that the expression is never equals to zero. Therefore, the polar curve does not have any horizontal tangent points.

To find the points where we have a vertical tangent, we must look for the points where the change in radius with respect to the angle is zero. From step 1, we found \(dr/d\theta\): $$\frac{dr}{d\theta} = 2\cos 2\theta$$ Set it equal to zero to find the vertical tangent points: $$2\cos 2\theta = 0$$ Solve for \(\theta\): $$2\theta = \pm\frac{\pi}{2} + n\pi \quad \text{where } n = 0, \pm 1, \pm 2, \dots$$ $$\theta = \pm \frac{\pi}{4} + \frac{n\pi}{2} \quad \text{where } n = 0, \pm 1, \pm 2, \dots$$ Now replace these values of \(\theta\) into the given function \(r = \sin(2\theta)\) to find the corresponding radius for each vertical tangent point: $$r = \sin\left(2\left(\pm \frac{\pi}{4} + \frac{n\pi}{2}\right)\right)$$ So the vertical tangent points are at \(r = \sin\left(2\left(\pm \frac{\pi}{4} + \frac{n\pi}{2}\right)\right)\), where \(n = 0, \pm 1, \pm 2, \dots\).

The polar curve \(r = \sin(2\theta)\) has no horizontal tangent points and vertical tangent points at \(r = \sin\left(2\left(\pm \frac{\pi}{4} + \frac{n\pi}{2}\right)\right)\), where \(\theta = \pm \frac{\pi}{4} + \frac{n\pi}{2}\) and \(n = 0, \pm 1, \pm 2, \dots\).