Questions: Sketch a graph of the polar equation and find the tangent lines at the pole (if any). r=2(1-cos(θ))

Solution Steps

The given polar equation is: \[ r = 2(1 - \cos(\theta)) \]

To find the tangent lines at the pole, we first convert the polar equation to Cartesian coordinates. Using the relationships \( r = \sqrt{x^2 + y^2} \) and \( \cos(\theta) = \frac{x}{r} \), we get: \[ r = 2(1 - \frac{x}{r}) \] Multiplying both sides by \( r \): \[ r^2 = 2r - 2x \] Substituting \( r^2 = x^2 + y^2 \): \[ x^2 + y^2 = 2\sqrt{x^2 + y^2} - 2x \]

Let \( r = \sqrt{x^2 + y^2} \): \[ x^2 + y^2 = 2r - 2x \] \[ r^2 = 2r - 2x \] \[ r^2 - 2r + 2x = 0 \]

At the pole, \( r = 0 \): \[ 0 = 2(1 - \cos(\theta)) \] \[ 1 - \cos(\theta) = 0 \] \[ \cos(\theta) = 1 \] Thus, \( \theta = 0 \) or \( \theta = 2\pi \). The tangent lines at the pole are horizontal lines.

Final Answer