How to solve the equation shown in image below?

Question

Megha 2020-9-8

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/590674-how-to-solve-the-equation-shown-in-image-below

评论： Image Analyst 2020-9-13

在 MATLAB Online 中打开

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!

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Answer 1

Image Analyst 2020-9-8

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/590674-how-to-solve-the-equation-shown-in-image-below#answer_491662

在 MATLAB Online 中打开

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);