Question

Verify that if the curve whose polar equation is \(r=f(\theta)\) is rotated about the pole through an angle \(\phi,\) then an equation for the rotated curve is \(r=f(\theta-\phi)\)

Short answer

Therefore, we can conclude that a rotation of the curve \(r = f(\theta)\) about its pole by an angle \(\phi\) leads to a new curve described by the equation \(r = f(\theta - \phi)\).

Step-by-step solution

Write out the problem statement

We need to prove that if the curve with polar equation is \(r = f(\theta)\), is rotated about the pole through an angle \(\phi\), then an equation for the rotated curve is \(r = f(\theta - \phi)\).

Expressing rotation using old polar coordinates

The polar coordinates of a point (r, \theta) on the original curve will become (r, \theta + \phi) once the curve is rotated counter-clockwise by an angle \phi.

Create new polar equation for rotated curve

So, the polar coordinates of a point on the new (rotated) curve is (r, \theta + \phi). In order to express r as a function of \theta, we substitute \(\theta + \phi\) with \(\theta\) and we get \(r = f(\theta)\). This means the equation for the rotated curve is \(r = f(\theta - \phi)\).