rewrite the given equation using the substitutions x=r cos theta and y=r sin theta. Simplify your answer. x^2+y^2+3x=0
so, what's the trouble? Just plug and chug.
x^2+y^2 = r^2, so
r^2+3rcosθ = 0
r(r+3cosθ) = 0
If you want just r=f(θ) then you have to solve for r using the quadratic formula.
To rewrite the equation using the substitutions x = r cos(theta) and y = r sin(theta), we substitute these expressions into the given equation.
Start with the equation:
x^2 + y^2 + 3x = 0
Replace x with r cos(theta) and y with r sin(theta):
(r cos(theta))^2 + (r sin(theta))^2 + 3(r cos(theta)) = 0
Simplify each term using trigonometric identities:
r^2 cos^2(theta) + r^2 sin^2(theta) + 3r cos(theta) = 0
Since cos^2(theta) + sin^2(theta) = 1, we can simplify the equation further:
r^2 + 3r cos(theta) = 0
Therefore, the simplified equation in terms of r and theta is:
r^2 + 3r cos(theta) = 0.