Convert 2x^2-2y^2=8y to polar form. Express as the equation r(theta).
ditch the factor of 2, and you have
r^2 cos^2θ - r^2 sin^2θ = 8 r sinθ
r^2 cos(2θ) = 8r sinθ
r cos(2θ) = 8 sinθ
r = 8sinθ/cos(2θ)
rats. Change all those 8's to 4's.
To convert the equation 2x^2 - 2y^2 = 8y to polar form, we need to express x and y in terms of r and 𝜃.
Step 1: Convert x and y to polar coordinates.
We know that x = r*cos(𝜃) and y = r*sin(𝜃).
Step 2: Substitute the values of x and y into the equation.
So, we have:
2(r*cos(𝜃))^2 - 2(r*sin(𝜃))^2 = 8(r*sin(𝜃))
Simplifying this, we get:
2r^2*cos^2(𝜃) - 2r^2*sin^2(𝜃) = 8r*sin(𝜃)
Step 3: Rearrange the equation.
We can rearrange this equation to isolate r:
2r^2*(cos^2(𝜃) - sin^2(𝜃)) = 8r*sin(𝜃)
2r^2*cos(2𝜃) = 8r*sin(𝜃)
Step 4: Divide both sides of the equation by 2r.
This will simplify the equation further:
r*cos(2𝜃) = 4*sin(𝜃)
Step 5: Convert the trigonometric functions using double-angle identities.
Using the double-angle identities for cosine and sine, we can rewrite the equation as:
r*(cos^2(𝜃) - sin^2(𝜃)) = 4*sin(𝜃)
r*cos^2(𝜃) - r*sin^2(𝜃) = 4*sin(𝜃)
Step 6: Rewrite the trigonometric expression with the Pythagorean identity.
Using the Pythagorean identity sin^2(𝜃) + cos^2(𝜃) = 1, we can substitute cos^2(𝜃) with 1 - sin^2(𝜃), getting:
r*(1 - sin^2(𝜃)) - r*sin^2(𝜃) = 4*sin(𝜃)
Simplifying further, we have:
r - r*sin^2(𝜃) - r*sin^2(𝜃) = 4*sin(𝜃)
r - 2r*sin^2(𝜃) = 4*sin(𝜃)
Step 7: Rearrange the equation and factor out r.
We can rearrange the equation to isolate r, and then factor it out:
r = 4*sin(𝜃) / (1 - 2sin^2(𝜃))
So, the equation in polar form, expressed as r(𝜃), is:
r(𝜃) = 4*sin(𝜃) / (1 - 2*sin^2(𝜃))