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I have the following cylindrical equation:
z = r^2 cos(2theta)
I am suppose to convert it into a rectangular equation. I'm stumped.
z = r^2 cos(2theta)
I am suppose to convert it into a rectangular equation. I'm stumped.
yeah i realized i could use cos(2theta) from the previous problem but im still stuck since i still have the 4xy on the other side of the equation :(Originally posted by matt grime
You seem to be doing well enough on your own:
you know what x^2-y^2 is from the previous example.
What other trig identities do you know? say, sin(2theta)?
yep thats what i got. thanksOriginally posted by HallsofIvy
Since I am not particularly bright and have great difficulty remembering trig identities, I would probably change
z^2 (x^2 - y^2) = 4xy into polar coordinates in the obvious way:
Since x= r sinθ and y= r cosθ, x^2= r^2 sin^2θ, y^2= r^2 cos^2θ so x^2- y^2= r^2(cos^2θ- sin^2θ), and 4xy= 4r^2 cosθ sinθ. Now the "r^2" terms cancel leaving z^2(cos^2θ-sin^2θ)= 4sinθcosθ or
z^2= (4sinθcosθ)/(cos^2θ- sin^2θ).
Now IF I were smart I might remember (or look up!) both
"cos(2θ)= cos^2θ- sin^2θ" and
"sin(2θ)= 2sinθcosθ to write that equation as
z^2= 2sin(2θ)/cos(2θ)