Elissa89 said:My professor gave us a study guide with the solutions:
The equation is:
sec(theta)=2
I am supposed to convert it to a rectangular equation. I know the answer is going to be y^2-3(x)^2=0
I don't know how to get to the answer he gave us.
skeeter said:note that $x = r\cos{\theta}$ and $r=\sqrt{x^2+y^2}$ ...$\sec{\theta} = \dfrac{1}{\cos{\theta}} = 2 \implies \cos{\theta}=\dfrac{1}{2}$
multiply both sides by $r$ ...
$r\cos{\theta} = \dfrac{r}{2} \implies x = \dfrac{\sqrt{x^2+y^2}}{2} \implies x^2 = \dfrac{x^2+y^2}{4}$$4x^2 = x^2+y^2 \implies y^2 - 3x^2 = 0$
A polar equation is a mathematical expression that relates the distance from the origin and the angle of a point in the polar coordinate system. It is written in the form of r = f(θ), where r represents the distance and θ represents the angle.
To convert a polar equation to a rectangular equation, we use the following formulas: x = r cos(θ) and y = r sin(θ). These formulas represent the coordinates of a point in the rectangular coordinate system based on its distance from the origin and its angle in the polar coordinate system.
The polar equation for secant is r = 1/cos(θ). This equation represents a curve that is symmetric about the x-axis and has vertical asymptotes at θ = (2n+1)π/2, where n is an integer.
To solve a polar equation for a specific value of θ, we substitute the value into the equation and solve for r. This will give us the distance of the point from the origin at that particular angle.
The rectangular equation for sec(θ) = 2 is x = 1/2 and y = 0. This represents a horizontal line passing through the point (1/2, 0) on the x-axis.