Question

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

Short 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-by-step solution

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.

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.

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 \(\frac{\pi}{5}\).

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.