Find a Cartesian equation for the curve and identify it.
(r^2)cos(2θ)=1
r^2(cos^2θ-sin^2θ) = 1
r^2 cos^2θ - r^2 sin^2θ = 1
x^2-y^2 = 1
To find the Cartesian equation for the curve, we start by using the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ):
x = r * cos(θ)
y = r * sin(θ)
Now, we need to express the given equation in terms of r, θ, x, and y. Let's rearrange the given equation:
(r^2) * cos(2θ) = 1
We can use the double angle formula for cosine to simplify:
cos(2θ) = cos^2(θ) - sin^2(θ)
Plugging this into the equation, we have:
(r^2) * (cos^2(θ) - sin^2(θ)) = 1
Let's substitute the values of x and y using the equations for converting from polar to Cartesian coordinates:
x = r * cos(θ)
y = r * sin(θ)
Substituting x and y into the equation, we get:
(x^2 - y^2) = 1
This is known as a Cartesian equation. Now, let's identify the curve.
The given equation, (r^2)cos(2θ) = 1, simplifies to (x^2 - y^2) = 1 in Cartesian coordinates. This equation represents a hyperbola with horizontal transverse axis centered at the origin.