Exercise7. (a)Sketchthecurvewithpolarequationr=2cosθ
(b)FindaCartesianequationforthiscurve.

### (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.

Explore how to sketch the polar curve r = 2cos(theta) and convert it to a Cartesian equation. Learn about polar and Cartesian coordinates, circle equations, and step-by-step solutions. Suitable for college-level mathematics.

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