As a continuation from the previous post, I am entering in and defining a function for tangential velocities. It is as easy as before. I set

u_theta2(u):= %pi*(lamb_da(al_pha) + csc(u)*cot(u) - (csc(u))^2 - log(tan(u/2)));

or

\[ u_{\theta} = \pi \left( \lambda + \csc \phi \cot \phi - \csc^2 \phi - \ln \Phi \right) \]

where \( \Phi = \tan \dfrac{\phi}{2} \).

Right now I am using the variable "u" in place of \( \phi \). I will try to change that to "phee" later.I will aslo try to make a function for "big_phee" for \( \Phi \) too.

I also create the function

**one_over_R_sin**(u) for comparison and choose a constant \( R \) while varying \( \phi \).

one_over_R_sin(u):= 1/(0.7*sin(u));

\[ u_{\theta} = \dfrac{1}{R \sin \phi} \]

Then we can plot by

plot2d([u_theta2(u), one_over_R_sin(u)], [u, 0.00001, %pi/6], [y, 0, 20]);

So the

**plot2d**a 2-d graph or plot. The first argument allow you to plot more than one function as the notation of

**[f(x), g(x)],**. If we were to just plot one function there would be no need for the square brace

**[ ]**but instead just the function

**f(x),**. The next argument allows the range for the "x" variable which in this case is

**u**. So I let

**u**range from 0 to my angle \(\alpha \) of \( 30^{\circ} \) or \( \dfrac{\pi}{6} \) radians. This must be encased in a square brace [ ]. Then, if we like, we can give a range for the "y-axis" which I ranged from 0 to 20.

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