Solving for angle (Beta) using vpasolve

Hello I'm trying to see why I'm not getting the correct angle when i Solve for the following equation. I'm not sure if this is due to a coding error that I dont see or if has to do with an angle property like the refence angle.

The equation I'm solving for is the following the resul i get is

beta =

-196.3722

However if im not mistaken the answer i should get is 23.13 deg.

M1 = 3;
P1 = 1;
theta2 = 5;
thetha3 = 20;
gamma = 1.4;

syms beta

tantheta = tand(theta2);

x = 2*cotd(beta)*((M1^2*((sind(beta))^2)-1));
y = M1^2*(gamma+cosd(2*beta))+2;
eqn = x/y;

result = vpasolve(eqn == tantheta,beta);

beta = result

Answer 1

[Star Strider]

That’s the result I get when I solve it (and then correct it by ) —

M1 = 3;
P1 = 1;
theta2 = 5;
thetha3 = 20;
gamma = 1.4;

syms beta

tantheta = tand(theta2);

x = 2*cotd(beta)*((M1^2*((sind(beta))^2)-1));
y = M1^2*(gamma+cosd(2*beta))+2;
eqn = x/y;

result = solve(tantheta == eqn, beta);

beta = vpa(result) + 180

beta = 

23.133257450785606135761291258498

.

[Benneth Perez]

Thanks, I'm not sure why but doing

result = vpasolve(tantheta == eqn, beta);

instead of result = solve(tantheta == eqn, beta);

was giving me a different answer.

[Star Strider]

As always, my pleasure!

The reason is that solve and vpasolve are different functions, and work differently.

The solve function solves the equation symbolically, and then the vpa call returns a numeric equivalent (although off from the desired result by 180°).

The vpasolve function is a numeric solver, and apparently does not solve symbolically first and then createes a numeric representation of that solution. There may be several different results, and since the function is not a poplynomial, only solves for one of them. Giving it different initial estimates for β may result in different results. See the Tips section for a discussion.

There are actually six solutions, so a different starting point would be necessary for each of them in order to return all of them with vpasolve —

M1 = 3;
P1 = 1;
theta2 = 5;
thetha3 = 20;
gamma = 1.4;
syms beta
tantheta = tand(theta2);
x = 2*cotd(beta)*((M1^2*((sind(beta))^2)-1));
y = M1^2*(gamma+cosd(2*beta))+2;
eqn = x/y - tantheta;

figure
hp = fplot(eqn, [0 359]);
grid
xv = hp.XData;
yv = hp.YData;
idx = find(diff(sign(yv)));
idx = idx([1:3 5:7]);
xv(idx);
for k = 1:numel(idx)
    idxrng = (-1:1)+idx(k);
    xexact(k) = interp1(yv(idxrng), xv(idxrng), 0);
end
xexact

xexact = 1×6

23.3458 88.1438 163.5677 203.2841 268.1513 343.5227

hold on
plot(xexact, zeros(size(xexact)), '+r')
hold off

I did this numerically because that is simply more efficient than making numerical guesses.

So vpasolve returned a ‘correct’ solution, simply not the desired solution!

.

[Benneth Perez]

Wow, thanks i figured i was off by 180 which in most cases worked but this starting points it was not working.

After noticing the code worked by just switching to solve i tried an input of theta2 = 15 and i started getting an odd answer again. Follow up I couldnt figure out what the issue was. I tried this new approach you have and it seems to work just fine. Although im wondering why my approach returns some weird solution.

I think part of it has to do with understanding the tangent plot and part of it with figuring out how to use vpasolve and solve. I appreciate your help! if you could take a quick look at my link that be great.

[Walter Roberson]

vpasolve() uses a modified Newton-Raphson algorithm. It might need to take the derivative of the function internally first.

[Star Strider]

As always, my pleasure!

I looked at the linked Question, and posted an Answer (of sorts) to it, although I could not find a symbolic solution for it.

.