Find a Cartesian Equation for the Curve and Identify it: r = 7 cos(8θ)

To convert the polar equation r = 7 cos(8θ) into a Cartesian equation, we start by using the relationships between polar and Cartesian coordinates. The standard transformations are:

• x = r cos(θ)
• y = r sin(θ)
• r² = x² + y²

From the given equation, we can multiply both sides by r (noting that this is valid as long as r is not zero):

r² = 7 r cos(8θ)

Now substituting for r²:

x² + y² = 7 r cos(8θ)

We need to express cos(8θ) in terms of x and y. Using the identity:

cos(θ) = x / r

This gives us:

cos(8θ) = cos(8 * arctan(y/x))

However, that’s a more complex transformation. Instead, we can simplify using the fact that:

r = sqrt(x² + y²)

To express cos(8θ), we note that:

r = 7cos(8θ)

Where θ = arctan(y/x). We could use de Moivre’s theorem to simplify cos(8θ). However, it’s simpler to express the form of the equation.

Ultimately, r = 7 cos(8θ) describes a rose curve, and the identification can be made without full Cartesian transformation:

• Type of Curve: Rose curve
• Number of Petals: 8 petals (since the coefficient of θ is 8)

In conclusion, although we can derive a complex Cartesian form, the original polar equation effectively describes a well-known rose curve, which is sufficient for most applications.