Failure to solve symbolic equations in the form 'Asin(x) + Bcos(x) == C'

Question

Gorkem 2018-3-7

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https://ww2.mathworks.cn/matlabcentral/answers/386908-failure-to-solve-symbolic-equations-in-the-form-a-sin-x-b-cos-x-c

回答： John D'Errico 2018-3-7

Left hand side of this equation is known to admit A*sin(x) + B*cos(x) = R*cos(x-alpha) with R = sqrt(A^2+B^2) and alpha = arctan(b/a). and x can be solved using this identity.

(see http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-rcostheta-alpha-2009-1.pdf for derivation).

However, MATLAB never finds the solution to x symbolically. It either looks for complex solutions (though I define everything to be real and positive) or returns really complicated expressions or returns an empty solution.

What's the best way to deal with these kind of equations?

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here is a sample code for the most general case:

syms a b c x real;
S = solve(a*sin(x) + b*cos(x) == c, x);

This returns an empty solution

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Walter Roberson 2018-3-7

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Answer 1

John D'Errico 2018-3-7

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I would point out that for SOME values (ok, many) of {a,b,c}, there is no real solution. So if you specify that all are real, including x, then what can you expect?

If abs(c)/sqrt(a^2 + b^2) is greater than 1, no real solution can exist.

I'll admit, I'd probably be lazy and just do the thinking for MATLAB here.

syms u
syms a b c real
S = solve(a*u + b*sqrt(1-u^2) == c,u)
S =
 (a*c + b*(a^2 + b^2 - c^2)^(1/2))/(a^2 + b^2)
 (a*c - b*(a^2 + b^2 - c^2)^(1/2))/(a^2 + b^2)

And we know that x = asin(u).

Sometimes symbolic solvers need to be gently coaxed down a reasonable path.