How to solve the equation shown in image below?

I wish to solve the eqn shown in figure below:

Here, I wish to find the value of theta_u. The i/p paarmeters are:

clear;
Vau = 43;
Vsu = 47.51; % 47.45
gamma = 5/3;
beta = 1.41;

I think I made some mathetcal error while taking theta_u on LHS-RHS. This leads me to wrong answer: i.e. 161.8754 (degree).

I coded it like:

syms theta_u
% I think there is some error in how I have solved equation and represented in q1
q1 = double(solve((acosd(Vau*cosd(theta_u)/Vsu)*sqrt(beta*gamma/2))/(theta_u)-1))

However, this caluculation goes wrong. The correct answer is 71.8 degree.

Could anyone help on mathematics and code?

Thank you in advance!

Answer 1

[Image Analyst]

I also get 161, as you can see by plotting it:

Vau = 43;
Vsu = 47.51; % 47.45
gamma = 5/3;
beta = 1.41;
theta_u = linspace(0, 360, 360);
y = acosd(Vau*cosd(theta_u)/Vsu)*sqrt(beta*gamma/2)
plot(theta_u, theta_u, 'r-', 'LineWidth', 2);
hold on;
plot(theta_u, y, 'b-', 'LineWidth', 2);
grid on;
% Find closest diff
[minDelta, index] = min(abs(y-theta_u))
crossingTheta = theta_u(index)
xline(crossingTheta, 'Color', 'k', 'LineWidth', 2);

Are you sure the gamma term is not supposed to be inside the acos()?

y = acosd((Vau*cosd(theta_u)/Vsu)*sqrt(beta*gamma/2))

but if I do that I get 90.2. Why do you say the answer is really 71.8?

[Image Analyst]

megha, please give an update.