Question

Find a cartesian equation for the curve r² cos(2θ) = 1. Identify the curve.

Answer 1

To find a Cartesian equation for the curve r² cos(2θ) = 1, we utilize the polar to Cartesian coordinates transformation and trigonometric identities. The resulting Cartesian equation is x² = y², which describes a circle centered at the origin with radius 1.

Step-by-step explanation:

To find a Cartesian equation for the curve given by r² cos(2θ) = 1, we can use the relationships between polar and Cartesian coordinates. In polar coordinates, x = r cos(θ) and y = r sin(θ). We can use a trigonometric identity to express cos(2θ) as 2cos²(θ) – 1. Substituting x/r for cos(θ), we have:

2x²/r² – 1 = 1/r²

After simplification and rearranging, we find:

x² = r²/2

Now, using r² = x² + y², we substitute for r² to get:

x² = (x² + y²)/2

And thus:

x² = y²

This is the equation of a circle centered at the origin with radius 1. The identification of the curve is a circle.