# Solving simultaneous trigonometric equations

Is it possible to solve these equations together ? PS:I am new to Mathematica

The only unknowns are theta 1 and alpha 1 , you can consider the rest constants I am using this

Reduce[{
y == ArcTan[(3000 - 30*Sin[2 x + 0.0093] - 30^2/(4*3000))/(30 +
30 Cos[2 x + 0.0093]),
x == (Pi/2 - 0.0093 - y)/2]
}, {x, y}]


but it returns an error requiring exact values

• Is a1 same as alpha1 ? May 1, 2017 at 5:42
• yes it is , sorry for that May 1, 2017 at 6:06
• There was a syntax issue with your code. Also, FindRoot is best if you are ok with numerical results. FindRoot[{y == ArcTan[(3000 - 30*Sin[2 x + 0.0093] - 30^2/(4*3000))/(30 + 30 Cos[2 x + 0.0093])], x == (Pi/2 - 0.0093 - y)/2}, {{x, 1}, {y, 1}}] May 1, 2017 at 6:19
• Thank you for the answer ! I just forgot to copy that syntax error fixed. May 1, 2017 at 6:43

Yes, the equations can be solved in MMA. Equation solving is an advanced subject. Here is one approach for solving the equations:

eqn = Tan[θ] == (f - r Sin[π/2 - θ] - z)/(x + r Cos[π/2 - θ])
eqn = eqn /. α -> (π/2 - θ - ω)/2
eqn = eqn /. f -> g + z
soln = Solve[eqn, θ] // Simplify

(*  {{θ -> ConditionalExpression[
ArcTan[(g*r - Sqrt[x^2*(g^2 - r^2 + x^2)])/(g^2 + x^2),
-((r*x^2 + g*Sqrt[x^2*(g^2 - r^2 + x^2)])/
(g^2*x + x^3))] + 2*Pi*C[1],
Element[C[1], Integers]]},
{θ -> ConditionalExpression[
ArcTan[(g*r + Sqrt[x^2*(g^2 - r^2 + x^2)])/(g^2 + x^2),
-((r*x^2 - g*Sqrt[x^2*(g^2 - r^2 + x^2)])/
(g^2*x + x^3))] + 2*Pi*C[1],
Element[C[1], Integers]]}}  *)


Explanation: In the first line we write our equation. Note the use of single = and double == signs. In the second line we replace all of $\alpha$'s with the expression we have. In the third line we replace the variable $f$ with $g+z$. In the fourth line, we solve for $\theta$.

Where did $g$ come from? First, we solved the equation without $g$ and got an even more complicated expression. We recognized that $f^2-2fz+z^2$ appeared several places in the solution and thought it would be simpler to write that part as some $g^2$. So, we tried it and it worked -- we did get a simpler expression. Try it without the third step and see which one you prefer.

We got 2 different expressions for the solution and each on them involves an arbitrary constant, C[1], which must be an integer. So there are an infinite number of solutions.

MMA has put the solution in terms of an arctangent function with 2 arguments. Those two arguments are $x, y$, not $y, x$.

• This (π/2 - θ - ω) should be (π/2 - θ - ω)/2?
– zhk
May 1, 2017 at 7:27
• @zhk Thank you! I have updated the code and the MMA solution. Good catch, Z. May 1, 2017 at 7:47