Question

Consider the following. x = tan²(θ), y = sec(θ), - π/2 <θ < π/2. Eliminate the parameter to find a cartesian equation that represents the curve.

Answer 1

To eliminate the parameter and find a Cartesian equation, we used trigonometric identities to express tan and sec in terms of sine and cosine. With substitution and algebraic manipulation, we derived the Cartesian equation x = y² - 1.

Step-by-step explanation:

To eliminate the parameter θ and find a Cartesian equation that represents the curve, we can use the given trigonometric identities.

Given x = tan²(θ) and y = sec(θ), and the fact that tan(θ) = sin(θ)/cos(θ) and sec(θ) = 1/cos(θ), we can write:

x = ²(sin(θ)/cos(θ))

y = 1/cos(θ)

Rearranging for cos(θ), we have:

cos(θ) = 1/y

Now, substitute cos(θ) in the expression for x:

x = sin²(θ) / cos²(θ)

x = sin²(θ) · y²

Since sin²(θ) + cos²(θ) = 1, we can write:

sin²(θ) = 1 - cos²(θ)

And because cos(θ) = 1/y, we have:

sin²(θ) = 1 - (1/y)²

Now, replace sin²(θ) in the expression for x:

x = (1 - (1/y)²) · y²

x = y² - 1

The Cartesian equation for the curve is therefore:

x = y² - 1