Ok, I have two questions first:
1. I'm asked to find a cartesian equation for the polar graph of this polar equation: r^2 = sin(2(theta))
The answer I got was
(x^2+y^2)^2/(2xy) = 2
Is this the correct way to express it?
2. I need to find the cartesian equation for this parametric equation:
x = sec(t) + tan(t)
y = sec(t) - tan(t)
I have tried using the identity tan^2x + 1 = sec^2x, but I feel like I'm moving in circles!
I'd appreciate it so much if someone could help me!!! Thanks =)
2. x + y = 2 sec t
x - y = 2 tan t
(1/4)(x+y)^2 = (1/4)(x-y)^2 + 1
This gets rid of the t terms. Rearrange and simplify.
1. To find the cartesian equation for a polar graph, you can substitute the given expression for r into the equation and then simplify.
For the polar equation r^2 = sin(2θ), we can substitute r^2 with (x^2+y^2) and sin(2θ) with 2sinθcosθ, because sin(2θ) = 2sinθcosθ.
So, we have (x^2+y^2) = 2sinθcosθ.
Now, to express this equation in terms of x and y only, we use the trigonometric identity sinθ = y/r and cosθ = x/r, where r is the distance from the origin to the point (x, y).
Substituting these identities into the equation, we get (x^2+y^2) = 2(y/r)(x/r).
To simplify further, we can multiply both sides of the equation by r^2 to get rid of the fractions, resulting in (x^2+y^2)^2 = 2xy.
Therefore, the correct cartesian equation for the given polar equation is (x^2+y^2)^2/(2xy) = 1.
2. To find the cartesian equation for a parametric equation, you can eliminate the parameter by expressing it in terms of x and y, and then simplifying the equation.
The given parametric equations are x = sec(t) + tan(t) and y = sec(t) - tan(t).
Let's start by adding the two equations:
x + y = (sec(t) + tan(t)) + (sec(t) - tan(t)) = 2sec(t).
Next, subtract the second equation from the first equation:
x - y = (sec(t) + tan(t)) - (sec(t) - tan(t)) = 2tan(t).
Now, using the identity tan^2(t) + 1 = sec^2(t), we can rewrite the equation 2tan(t) as 2(sec^2(t) - 1).
Combining the equations x + y = 2sec(t) and x - y = 2(sec^2(t) - 1), we can eliminate the parameter t:
(1/4)(x + y)^2 = (1/4)(x - y)^2 + 1.
Rearrange and simplify further if needed to obtain the final cartesian equation.