Rewrite the given equation using the substitutions x=rcos(theta) and y=rsin(theta). Simplify your answer.
To rewrite the given equation using the substitutions x = rcos(theta) and y = rsin(theta), we need to substitute these expressions in place of x and y in the equation and simplify.
Let's say the given equation is:
f(x, y) = 3x^2 + 2xy - y^2
Using the substitutions x = rcos(theta) and y = rsin(theta), we can rewrite this equation as follows:
f(rcos(theta), rsin(theta)) = 3(rcos(theta))^2 + 2(rcos(theta))(rsin(theta)) - (rsin(theta))^2
Now, let's simplify this equation:
f(rcos(theta), rsin(theta)) = 3r^2cos^2(theta) + 2r^2cos(theta)sin(theta) - r^2sin^2(theta)
Next, we can use trigonometric identities to simplify further. The identity cos^2(theta) - sin^2(theta) = cos(2theta) can be applied here:
f(rcos(theta), rsin(theta)) = 3r^2cos^2(theta) - r^2sin^2(theta) + 2r^2cos(theta)sin(theta)
Now, we can factor out r^2 from the first two terms and rearrange them as a difference of squares:
f(rcos(theta), rsin(theta)) = r^2(3cos^2(theta) - sin^2(theta)) + 2r^2cos(theta)sin(theta)
Finally, we simplify this further by using another trigonometric identity sin(2theta) = 2sin(theta)cos(theta):
f(rcos(theta), rsin(theta)) = r^2(3cos^2(theta) - sin^2(theta)) + 2r^2sin(theta)cos(theta)
And that is the simplified equation obtained by using the given substitutions.