What shapes are described with $\rho = \cos{(\phi)}$ and $\rho = \cos{(2\theta)}$?

$\begingroup$

I have started an Multivariable course, and I'm learning about spherical coordinates. My problem now is learn how to graph this kind of shapes.

This is the problem:

What shapes are described when...?

Solution:

a) $\rho = 1$ : Sphere with radius 1.

b) $\phi = \frac{\pi}{3}$ : Cone with angle $\frac{\pi}{3}$.

c) $\theta = \frac{\pi}{4}$ : Semi-circular cross-section with diameter along z-axis

d) $\rho = \cos{(\phi)}$ : ?

e) $\rho = \cos{(2\theta)}$ : ...?

Are they correct? How to describe, verbally, the last two -d) and e).

$\endgroup$ 5

1 Answer

$\begingroup$

If $\phi$ is a cone with angle $\pi/3$ then:

d) $\rho=\cos\phi$:

Multiply both terms by $\rho$ and you get $$ \rho^2=\rho\cos\phi \quad \Rightarrow \quad x^2+y^2+z^2=z, $$ which is a sphere of radius $1/2$ centered at $(0,0,1/2)$:

e) $\rho=\cos2\theta$:

We could eventually find the cartesian equation here, but it will not be of any help, as it is not a classical surface:

a), b) and c) are correct. To convince yourself, find the cartesian equations.

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy