Problem 25 - Essential Calculus

Question

Find all points (if any) of horizontal and vertical tangency to the portion of the curve shown. $$ \begin{aligned} &\text { Involute of a circle: }\\\ &x=\cos \theta+\theta \sin \theta\\\ &y=\sin \theta-\theta \cos \theta \end{aligned} $$

Step-by-Step Solution

Answer

The points of horizontal and vertical tangency to the curve are (1,0), (0,0), (0,\( \pm 1.89549426703298\)), (\( \pm 1.89549426703298\), 0), and (\( \pm 1.89549426703298\), \( \pm 1.89549426703298\)).

Step 1: Compute the derivatives of the parametric equations

The first step is finding the derivatives of the parametric equations \(x(\theta)\) and \(y(\theta)\) using calculus. The derivative of \(x(\theta) = \cos(\theta) + \theta \sin(\theta)\) with respect to \(\theta\) is \(x'(\theta) = -\sin(\theta) + \sin(\theta) + \theta \cos(\theta) = \theta \cos(\theta)\). Similarly, the derivative of \(y(\theta) = \sin(\theta) - \theta \cos(\theta)\) with respect to \(\theta\) is \(y'(\theta) = \cos(\theta) + \cos(\theta) - \theta \sin(\theta) = 2\cos(\theta) - \theta \sin(\theta)\).

Step 2: Find the values of \(\theta\) where the derivatives equal 0

We now set \(x'(\theta)\) and \(y'(\theta)\) equal to 0 and solve for \(\theta\). This will give us the points of tangency. Solving for \(\theta \cos(\theta) = 0\), we get \(\theta = 0\) or \(\cos(\theta) = 0\). The solutions to \(\cos(\theta) = 0\) are \(\theta = (\pi / 2) + n \pi\), where n is an integer. Solving \(2\cos(\theta) - \theta \sin(\theta) = 0\), we get \(\theta = 0\) or \(\cos(\theta) = \theta \sin(\theta) / 2\). This has an infinite number of solutions, so we'll only get the first few: \(\theta = 0\), and approximately \(\theta = 1.89549426703298\), and \(\theta = -1.89549426703298\).

Step 3: Substitute the values of \(\theta\) into the parametric equations to get the points of tangency

The last step is to substitute the values of \(\theta\) we found into the original parametric equations \(x(\theta)\) and \(y(\theta)\) to get the points of tangency. Plugging \(\theta = 0\) into \(x(\theta) = \cos(\theta) + \theta \sin(\theta)\) and \(y(\theta) = \sin(\theta) - \theta \cos(\theta)\) we get the point (1,0). Doing the same for the other values we found, we get the points \((0, 0)\), \((0, \pm 1.89549426703298)\), \((\pm 1.89549426703298, 0)\), and \((\pm 1.89549426703298, \pm 1.89549426703298)\).