Problem 64 - Essential Calculus

Question

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=-\sin 5 \theta $$

Step-by-Step Solution

Answer

The tangents at the pole can be drawn at points corresponding to the angles where the sign of \(r\) changes from negative to positive or vice-versa. The graph of the polar equation consists of a series of loops: five petals pointing in opposite directions.

Step 1: Sketching the graph

To start, plot the graph of the polar equation \(r = -\sin 5 \theta\). To do this, identify some key points by plugging in values for \(\theta\) into the equation and solving for \(r\). This will help create a table of values to use as a guide while sketching the graph.

Step 2: Interpreting the negative radius

In polar coordinates, if the radius \(r\) is negative, it simply means that the point lies in the opposite direction. After plotting the positive radii, mirror-image those points across the pole to account for the negative radii depicted in the polar equation.

Step 3: Tracing the curve

Once all key points are plotted, trace the curve using those points. The spikes of the graph occur at the origin; curve goes to the origin as \(\theta\) increases in increments of \(\frac{\pi}{5}\).

Step 4: Identifying locations for the tangents at the pole

Tangents to the polar graph at the pole occur when \(r\) changes from negative to positive or vice-versa. Identify the values of \(\theta\) where \(r\) changes sign. Then, draw the tangents at these points.