Problem 65 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The graph of the given polar equation is a circle centered at (0, 3/2) with a radius of 3/2. The tangent lines to this graph at the pole are the y-axis (x=0) and the x-axis (y=0).

Step 1: Convert Polar Equation to Cartesian Coordinates

A polar graph is different from a conventional Cartesian graph, it's based on a radius (r) and an angle (\(\theta\)). The conversion of the given polar equation \(r = 3 \sin 2\theta\) to Cartesian coordinates comes from the sine double angle formula. Thus, the corresponding Cartesian equation is \(x^2+ y^2 = 3y\).

Step 2: Sketch the Graph of the Equation

The equation can be rewritten as \(x^2 + y^2 - 3y = 0\), or, completing the square, \(x^2 + (y-3/2)^2 - (3/2)^2 = 0\). This represents a circle centered at (0, 3/2) with radius 3/2. Graph this circle on a Cartesian plane.

Step 3: Finding Tangent Lines at the Pole

In polar coordinates, the pole corresponds to the origin in Cartesian coordinates, (0,0). We are looking for lines tangent to the circle at the origin, which means the lines must pass through the origin and not intersect the circle again. The only lines that can do this are vertical and horizontal lines. The vertical line is the y-axis, and the horizontal line is the x-axis. Therefore, the tangents to the graph at the pole are x=0 and y=0.