I'm not going to put together a solution, but here's some code to get you thinking about how to do this.
>> alpha = linspace(0,2*pi);
>> cl = 2*sin(alpha).^2.*cos(alpha);
>> lower = cl < -0.525;
>> idx = find(lower, 1);
>> clBracket = [cl(idx-1) cl(idx)]
clBracket =
-0.4951 -0.58416
>> alphaBracket = [alpha(idx-1) alpha(idx)]
alphaBracket =
1.8405 1.904
You can see above that the CL value you specified is between the two CL "bracket" values. And the corresponding alphas are also shown. So, at this stage we know that alpha is between 1.8405 radians and 1.904 radians. The mean of these two values is a first-level approximation of alpha. You could follow this up with additional code that gets you closer to alpha. For example, you can repeat the statements above, except beginning with
alpha = linspace(alphaBracket(1), alphaBracket(2))
This will get you much tigher bracket values, and get you closer to an acceptable solution. Are you with me?
You can put all this in a loop and compute
abs(alphaBracket(1) - alphaBracket(2))
as the last step. When this value is below some acceptable value, usually called episilon, you're done.
Good luck.
