Problem 66 - Essential Calculus

Question

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=3 \cos 2 \theta $$

Step-by-Step Solution

Answer

The graph of the polar equation \(r=3 \cos 2 \theta\) is a rose with 4 petals. The tangent lines at the pole (origin) of this graph can be found by first converting it to cartesian coordinates, differentiating to find the derivative (slope at any point), then evaluating this derivative at the pole.

Step 1: Plotting the Polar Graph

We can interpret the equation \(r=3 \cos 2 \theta\) as a rose with 4 petals (twice the coefficient of theta). Also, each petal extends from the origin (pole) to \(r=3\) (amplitude of the cosine function). Plot points of this function starting from angle theta = 0 and moving in increments (like π/4) till we cover a full circle. Since the cosine function oscillates, we will end up tracing all the petals in one complete rotation.

Step 2: Conversion to Cartesian Coordinates

To find the tangent lines, it will be helpful to translate the equation to cartesian coordinates. We use the relationships: \(x = r\cos \theta\), \(y = r\sin \theta\), and \(r^2 = x^2 + y^2\). Therefore, substituting for \(r\) into \(x = r\cos \theta\) and \(y = r\sin \theta\) gives \(x = 3\cos 2\theta \cdot \cos \theta\) and \(y= 3\cos 2\theta \cdot \sin \theta\). These can be further reduced using double-angle formulae to \(x = 3(1 - 2\sin^2 \theta)\) and \(y = 3(2\sin \theta \cos \theta)\) respectively.

Step 3: Differentiating and Finding Tangent Lines

First, find the derivative of \(y\) with respect to \(x\) by implicit differentiation, this will give us the slope of the tangent at any point. At the pole (origin), \(x = y = 0\). Therefore, substitute \(\theta=0\) (note we set \(\theta\), not \(x\) or \(y\), to zero because this sets both \(x\) and \(y\) to zero) into the derivative to get the slopes for the tangent lines at the pole. Intersect these lines with the graph for the visual understanding.