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Chapter 8: Problem 56

Find the points of horizontal tangency (if any) to the polar curve. $$ r=a \sin \theta \cos ^{2} \theta $$

Short Answer

Expert verified
The exact solution depends highly on the value of 'a' and may involve complex trigonometric solutions. The solution would involve handling various trigonometric identities and could involve multiple values of \(\theta\).

Step by step solution

01

Compute the derivative of r with respect to \(\theta\)

First, we compute the derivative of \( r \) with respect to \(\theta\). The given equation is \( r = a \sin \theta \cos^2 \theta \), so applying the product and chain rules for differentiation gives \(\frac{dr}{d\theta} = a \cos \theta \cos^2 \theta - 2a \sin^2 \theta \cos \theta \).
02

Set the slope to zero

The slope of the tangent line in polar coordinates is given by \( \frac{dr}{d\theta}/ (1 - r \sin \theta) \). For horizontal tangents, this has to be zero. We solve \(a \cos \theta \cos^2 \theta - 2a \sin^2 \theta \cos \theta / (1 - a \sin \theta \cos^2 \theta \sin \theta) = 0\) for \(\theta\).
03

Solve the equation

Solving the equation will give us the values of theta at which the slope of the tangent line to the polar curve is zero, thus identifying the points of horizontal tangency. Factor out common terms where possible, and find values of \(\theta\) that satisfy the equation. Note that some values may not yield real points on the curve, thus should be disregarded.

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Most popular questions from this chapter

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Cycloid: } x=2(\theta-\sin \theta), \quad y=2(1-\cos \theta) $$

Give the integral formulas for area and arc length in polar coordinates.

A curve called the folium of Descartes can be represented by the parametric equations \(x=\frac{3 t}{1+t^{3}} \quad\) and \(y=\frac{3 t^{2}}{1+t^{3}}\) (a) Convert the parametric equations to polar form. (b) Sketch the graph of the polar equation from part (a). (c) Use a graphing utility to approximate the area enclosed by the loop of the curve.

In Exercises 47 and 48, use the integration capabilities of a graphing utility to approximate to two decimal places the area of the surface formed by revolving the curve about the polar axis. $$ r=4 \cos 2 \theta, \quad 0 \leq \theta \leq \frac{\pi}{4} $$

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=4 \sin 2 \theta, y=2 \cos 2 \theta $$
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