Polar to Rectangular conversions

[Eng67]

I am at a standstill with the solution to this problem.

I need to convert r^2=2cos(2 theta) to rectangular form.

I know that x = rcos(theta) and y = rsin(theta)

so far I have r = (2cos(2theta))/r

then I substitute for r

sqrt(x^2+y^2)= (2cos(2theta))/sqrt(x^2+y^2)

Then I hit a brick wall.

please help me knock down this wall.

Thanks

 

[TD]

Replace r² by x²+y² and theta = arctan(y/x).

 

[Eng67]

So I then would have

sqrt(x^2+y^2)= (2cos(2arctan y/x))/sqrt(x^2+y^2)

I am still stuck.

 

[HallsofIvy]

I am at a standstill with the solution to this problem.

I need to convert r^2=2cos(2 theta) to rectangular form.

I know that x = rcos(theta) and y = rsin(theta)

so far I have r = (2cos(2theta))/r

then I substitute for r

sqrt(x^2+y^2)= (2cos(2theta))/sqrt(x^2+y^2)

Then I hit a brick wall.

please help me knock down this wall.

Thanks

Back up a little! You have $r^2= 2 cos(2\theta)$ so first note that $cos(2\theta)= cos^2(\theta)- sin^2(\theta)$ so that
$r^2= 2(cos^2(\theta)- sin^2(\theta))$
Now multiply on both sides by r² to get
$(r^2)^2= 2(r^2cos^2(\theta)- r^2sin^2(\theta))$
I'll bet you can convert that to rectangular coordinates!

 

[Eng67]

Thanks!

This is now so simple.