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