(a) Sketch the curve with polar equation $r = 2\cos\theta$

The polar equation $r = 2\cos\theta$ represents a circle. In polar coordinates, $r$ is the distance from the origin to a point on the curve, and $\theta$ is the angle that the line from the origin to the point makes with the positive $x$-axis.

To sketch the curve:

1. Identify the type of curve: The given equation is of the form $r = a\cos\theta$, which is known to be a circle in polar coordinates.

2. Determine the radius and center:

   • The maximum value of $r$ occurs when $\cos\theta = 1$, giving $r = 2$.
   • The minimum value of $r$ occurs when $\cos\theta = -1$, giving $r = -2$, which reflects the curve across the origin.
   • The curve is a circle with a radius of $1$ centered at $(1, 0)$ in Cartesian coordinates.

3. Plot key points:

   • At $\theta = 0$, $r = 2$.
   • At $\theta = \frac{\pi}{2}$, $r = 0$.
   • At $\theta = \pi$, $r = -2$, which corresponds to $r = 2$ and $\theta = 0$.
   • At $\theta = \frac{3\pi}{2}$, $r = 0$.

The curve is a circle that is symmetric about the $x$-axis.

(b) Find a Cartesian equation for this curve

To convert the polar equation $r = 2\cos\theta$ into Cartesian coordinates, use the relationships between polar and Cartesian coordinates: $x = r\cos\theta \quad \text{and} \quad y = r\sin\theta$ $r^2 = x^2 + y^2$ From the polar equation $r = 2\cos\theta$, multiply both sides by $r$ to get: $r^2 = 2r\cos\theta$ Substitute $r^2 = x^2 + y^2$ and $r\cos\theta = x$ into the equation: $x^2 + y^2 = 2x$ Rearrange the equation: $x^2 - 2x + y^2 = 0$ Complete the square for $x$: $(x - 1)^2 - 1 + y^2 = 0$ $(x - 1)^2 + y^2 = 1$ This is the equation of a circle with a radius of $1$ centered at $(1, 0)$ in Cartesian coordinates.

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Do you want more details or have any questions?

Here are five related questions to explore further:

1. How would you sketch the polar curve for $r = 2\sin\theta$?
2. What is the Cartesian equation for the polar curve $r = 3\cos\theta$?
3. How can you derive the equation of an ellipse from a given polar equation?
4. What is the significance of completing the square when converting from polar to Cartesian coordinates?
5. How does the graph of $r = 2\cos\theta$ change if we replace the 2 with another constant?

Tip: When converting from polar to Cartesian coordinates, remember to use the identities $x = r\cos\theta$ and $y = r\sin\theta$ to simplify the process.