# Solving large trigonometric fucntions

Hi everyone,

I'm new to sage and would like to use it to give me an exact solution to the derivative of a large trig function. I have been tinkering around with the different trig simplification routines to no avail. Here is the current incarnation of my code:

var("theta phi r I sigma_a1 sigma_a2 M_1 M_2 F_tu")
import sympy as sy
M1(phi)=M_1*cos(phi+pi/2)
M1=M1.trig_reduce()
M2(phi)=M_2*cos(phi+theta+pi/2)
M2=M2.trig_reduce()
sigma_b1(r,phi)=M1(phi - pi/2)*r/I+sigma_a1
sigma_b1=sigma_b1.trig_reduce()
sigma_b2(r,phi)=M2(phi - pi/2)*r/I+sigma_a2
sigma_b2=sigma_b2.trig_reduce()
sigma_alt(r,phi)=sigma_b1-sigma_b2
sigma_alt=sigma_alt.trig_reduce()
sigma_mean(r,phi)=(sigma_b1-sigma_b2)/2
sigma_mean=sigma_mean.trig_reduce()
sigma_alteq(r,phi)=F_tu*sigma_alt/(F_tu-sigma_mean)
sigma_alteq=sigma_alteq.trig_reduce()
sol=diff(sigma_alteq,phi)
sol_reduc=SR(sy.fu(sy.sympify(sol))); sol_reduc
show(sol_reduc)
solve(sol_reduc,phi)


Sorry, I do not know how to make the code appear in a window. The final result it gives me is:

[sin(phi + theta) == M_1*sin(phi)/M_2]

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Could you please tell us which result would you expect ?

@Kevin_Bryant

To display blocks of code, either indent them with 4 spaces, or select the corresponding lines and click the "code" button (the icon with '101 010').

Can you edit your question to do that?

@tmonteil I was expecting a set of 4 solutions over the interval of 0 to 2*pi for phi. I solved this problem earlier on Wolframalpha before I added the terms sigma_a1 and sigma_a2 to my equations and got the following for one solution (note that variable names are different but the problem was essentially the same)

t = -cos^(-1)((-sqrt((2 F X Y^2 Z cos(2 a) - 2 F X Y^2 Z)^2 - 4 (-4 F^2 X Y cos(a) + 2 F^2 X^2 + 2 F^2 Y^2) (4 F^2 X Y cos(a) - F^2 Y^2 cos(2 a) - X^2 Y^2 Z^2 cos(2 a) - 2 F^2 X^2 - F^2 Y^2 + X^2 Y^2 Z^2)) - 2 F X Y^2 Z cos(2 a) + 2 F X Y^2 Z)/(2 (-4 ...(more)

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It seems to me that the hard work is already done properly by the solver. Finally, you got:

sage: var("theta phi M_1 M_2")
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)


Which is: $$\newcommand{\Bold}{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right)$$

Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$, recall that: • if $\theta$ is an integer multiple of $\pi$, then ${0,\pm \pi}$ is the solution set.
• in general, there are exactly $2$ intersections (and the solutions differ by $\pi$).
• if the values of $M_1$ and $M_2$ allow it ($|M_1/M_2| \leq 1$), then there exists a value of $\theta$ for which the denominator vanishes, but the solution set is $\pm \pi/2$.

Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with SymPy solve or the new solveset.. but maybe I'm missing to pass some extra information.

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