To convert the polar equation r(θ) = 2secθ into parametric equations, we can make use of the fundamental relationships between polar and Cartesian coordinates.
Step 1: Convert Polar Coordinates to Cartesian Coordinates:
The Cartesian coordinates (x, y) can be obtained by using the following formulas:
x = r(θ) * cos(θ)
y = r(θ) * sin(θ)
In this case, we have:
x = 2secθ * cos(θ)
y = 2secθ * sin(θ)
Step 2: Manipulate the Equations:
Since sec(θ) = 1/cos(θ), we can rewrite the equations as:
x = 2/cos(θ) * cos(θ)
y = 2/cos(θ) * sin(θ)
Simplifying further:
x = 2
y = 2 * tan(θ)
Step 3: Convert the Cartesian Coordinates to Parametric Equations:
To convert the Cartesian equations to parametric equations, we can express x and y in terms of a third variable, usually denoted as t.
Let's consider t = θ for our parametric equations:
x = 2
y = 2 * tan(t)
Thus, the parametric equations are:
x = 2
y = 2tan(t)
That's it! The polar equation r(θ) = 2secθ has been converted into parametric equations x = 2 and y = 2tan(t).