Convert the given polar equation to a Cartesian equation.
r^2=4sec(theta)csc(theta)
secØ = r/x , cscØ = r/y
so r^2 = 4(r/x)(r/y)
r^2 = 4r^2/(xy)
1 = 4/(xy)
xy = 4
Well, well, well, looks like we have a polar equation that wants to be transformed into a Cartesian equation. Let's dive in and do some circus mathematics!
To convert this equation, we'll make use of the following trigonometric identities:
- sec(theta) = 1/cos(theta)
- csc(theta) = 1/sin(theta)
Now, let's substitute these identities and see what happens:
r^2 = 4 * (1/cos(theta)) * (1/sin(theta))
Simplifying further, we get:
r^2 = 4/(cos(theta) * sin(theta))
Congratulations, my dear friend! We have successfully converted the polar equation to a Cartesian equation.
To convert the given polar equation to a Cartesian equation, we can use the substitution:
x = r*cos(theta)
y = r*sin(theta)
Let's apply this substitution to "r^2 = 4*sec(theta)*csc(theta)":
x^2 + y^2 = 4*(1/cos(theta))*(1/sin(theta))
Simplifying further:
x^2 + y^2 = 4/(cos(theta)*sin(theta))
Now, we need to eliminate the trigonometric functions in terms of x and y. To do that, we can use the following identities:
cos(theta) = x/r
sin(theta) = y/r
Substituting these identities into the equation, we get:
x^2 + y^2 = 4/(x*y)
Therefore, the Cartesian equation equivalent to the given polar equation "r^2 = 4*sec(theta)*csc(theta)" is:
x^2 + y^2 = 4/(x*y)
To convert the given polar equation to a Cartesian equation, we need to express the equation in terms of x and y variables.
First, recall the following equations relating polar and Cartesian coordinates:
x = r * cos(theta)
y = r * sin(theta)
Now, let's substitute these equations into the given polar equation:
r^2 = 4sec(theta) * csc(theta)
Substituting r^2 with (x^2 + y^2):
x^2 + y^2 = 4sec(theta) * csc(theta)
Next, let's express sec(theta) and csc(theta) in terms of sine and cosine:
sec(theta) = 1 / cos(theta)
csc(theta) = 1 / sin(theta)
Substituting these expressions into the equation:
x^2 + y^2 = 4(1 / cos(theta))(1 / sin(theta))
x^2 + y^2 = 4 / (cos(theta) * sin(theta))
Now, let's express cos(theta) and sin(theta) in terms of x and y:
cos(theta) = x / r
sin(theta) = y / r
Substituting these expressions into the equation:
x^2 + y^2 = 4 / ((x / r) * (y / r))
x^2 + y^2 = 4r / (x * y)
Finally, using the equation r = sqrt(x^2 + y^2):
x^2 + y^2 = 4(sqrt(x^2 + y^2)) / (x * y)
This is the Cartesian equation that is equivalent to the given polar equation.