Analytic Solutions to a Few Trig Equations

[Dissident Dan]

Is there an analytic solution to an equation of the following form?

A*cos(w*t) + B*t = C

where A, B, C, and w are constants

Maybe it can be solved by expanding the cos() to a series?

I am also wondering the same question about the following, though I believe that I've read/been told that there is no known analytic solution.

A*cos($$\Theta$$) + B*sin($$\Theta$$) = C

 

[gunch]

If what you mean by an analytical solution is a finite expression using only "elementary functions" then I don't believe the first has an analytical solution. You could of course always define a solution set:
$$S = \{t | A\cdot \cos(wt) + Bt = C\}$$
which I would consider a solution, though it doesn't tell us how to solve it.

For the second it's pretty easy. Rearrange:
$$A \cdot \cos(\Theta) = C - B\cdot \sin (\Theta)$$
Square:
$$A^2 (1-\sin^2(\Theta)) = C^2 + B^2 \sin^2(\Theta) - 2BC\cdot \sin(\Theta)$$
Then it's a simple quadratic equation in $$\sin(\Theta)$$.

 

[Dissident Dan]

Thanks!

The solution to the second is so simple, I almost can't believe I didn't come up with it. I guess that shows what happens when you haven't had a math class in a few years.

By analytic solution, I mean an equation solved for t, instead of a numerical method.

 

[Defennder]

Don't think there is a closed form solution. You'll have to approximate it numerically.