Find out θ in the cos(wt+θ) = A

Hello and thank you for reading my question,

Following problem: I got 2 sinusidal waves. One wave(f1,red) starts at point t1 and ends at point t2. The other(f2,blue) starts at point t2 and ends at point t3. I am trying to find out based on the difference in the amplitude at point t2 the θ-angle between those functions in order to get function f3(green), which will be a reconstruction of function f2. Both functions are normalized, with Amax = [1,-1] and both f1 and f2 are functions of the type cos(wt+θ); where for f1, θ=0 and for f2 θ is searched.

Now I tried to solve this with following code:

syms theta

A = f2(1);
tau = t12(end)-t12(1); % center red function so that t1 is 0
w = 2*pi*rpm;
eqn2 = cos(w*tau + theta) == A;
eqn2 = solve(eqn2,theta);

Which makes sense, because if I do this in my calculator I get many answers and the one near 0 is the right one. Thats why I could build the green function. But Matlab seems to give me only 2 answers for theta and I don't get why... Is there a way to solve this problem? Or am I doing something wrong?

Thank you for your help!

Answer 1

[MULI]

Hi Issac,

I understand that when the equation is solved using "solve" function it is providing two solutions.

Since cosine function has period 2*pi there are infinitely many solutions to the equation. MATLAB typically returns the principal values that are in range [-pi, pi] which satisfies the equation. Due to this reason, it provides two values.

Hope this answer clarifies your query!

Answer 2

[Walter Roberson]

eqn2 = solve(eqn2, theta, 'returnconditions', true);

will probably return a solution that is parameterized in terms of an extra variable, with the conditions returned describing the relationship that has to be satisfied for the extra variable.