Question

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

Short answer

The graph of our polar equation, \(r = 3 \sin \theta\), is a circle of radius 3 centered at the origin. This circle is traversed twice as \( \theta\) ranges from 0 to \( 2\pi\). The tangent lines at the pole, or origin, are the x-axis and negative x-axis.

Step-by-step solution

Understanding the Polar Equation

First, let's try to get a feel for our function. The given polar equation is \(r = 3 \sin \theta\). This describes a circle of radius 3 centered at the origin in the polar coordinate system. It is important to note that for \( \theta = 0 \) or \( \theta = \pi \), we have \( r = 0 \) or \( r = 3 \), respectively.

Sketching the Polar Graph

The nature of the sin function means that our circle is traversed as \(\theta\) ranges from 0 to \( \pi\), then backtracked (traced out again but in reverse order) as \(\theta\) ranges from \( \pi\) to \(2 \pi\). Therefore, we sketch a circle with radius 3 in the polar coordinate system. Please note that a negative value for r does not alter its direction but it changes the location of the point(pole).

Find the Tangents at the Pole

In general, if an equation of a curve in polar coordinates is given by \( r = f(\theta) \), then the polar tangents are given when \( dr/d\theta = 0 \). Differentiating \( r = 3 \sin \theta\) with respect to \(\theta\) gives \( dr/d\theta = 3 \cos \theta \). Setting this equal to zero gives \( \theta = \pi/2, 3\pi/2 \). These are the angles at which the tangents to the polar graph occur at the pole. In our case, this will be the x-axis and the negative x-axis, forming two tangents at the origin.