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sin(pi*(x-3)/4)=5*pi/4 equation

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Numerical solution:

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The solution

You have entered [src]
   /pi*(x - 3)\   5*pi
sin|----------| = ----
   \    4     /    4  
$$\sin{\left(\frac{\pi \left(x - 3\right)}{4} \right)} = \frac{5 \pi}{4}$$
Detail solution
Given the equation
$$\sin{\left(\frac{\pi \left(x - 3\right)}{4} \right)} = \frac{5 \pi}{4}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -1

The equation is transformed to
$$\sin{\left(\frac{\pi x}{4} + \frac{\pi}{4} \right)} = - \frac{5 \pi}{4}$$
As right part of the equation
modulo =
True

but sin
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
The graph
Rapid solution [src]
                /    /5*pi\\         /    /5*pi\\
       pi + 4*re|asin|----||   4*I*im|asin|----||
                \    \ 4  //         \    \ 4  //
x1 = - --------------------- - ------------------
                 pi                    pi        
$$x_{1} = - \frac{\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} - \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}$$
             /    /5*pi\\         /    /5*pi\\
         4*re|asin|----||   4*I*im|asin|----||
             \    \ 4  //         \    \ 4  //
x2 = 3 + ---------------- + ------------------
                pi                  pi        
$$x_{2} = \frac{4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + 3 + \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}$$
x2 = 4*re(asin(5*pi/4))/pi + 3 + 4*i*im(asin(5*pi/4))/pi
Sum and product of roots [src]
sum
           /    /5*pi\\         /    /5*pi\\           /    /5*pi\\         /    /5*pi\\
  pi + 4*re|asin|----||   4*I*im|asin|----||       4*re|asin|----||   4*I*im|asin|----||
           \    \ 4  //         \    \ 4  //           \    \ 4  //         \    \ 4  //
- --------------------- - ------------------ + 3 + ---------------- + ------------------
            pi                    pi                      pi                  pi        
$$\left(\frac{4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + 3 + \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}\right) + \left(- \frac{\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} - \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}\right)$$
=
             /    /5*pi\\       /    /5*pi\\
    pi + 4*re|asin|----||   4*re|asin|----||
             \    \ 4  //       \    \ 4  //
3 - --------------------- + ----------------
              pi                   pi       
$$- \frac{\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + \frac{4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + 3$$
product
/           /    /5*pi\\         /    /5*pi\\\ /        /    /5*pi\\         /    /5*pi\\\
|  pi + 4*re|asin|----||   4*I*im|asin|----||| |    4*re|asin|----||   4*I*im|asin|----|||
|           \    \ 4  //         \    \ 4  //| |        \    \ 4  //         \    \ 4  //|
|- --------------------- - ------------------|*|3 + ---------------- + ------------------|
\            pi                    pi        / \           pi                  pi        /
$$\left(- \frac{\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} - \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}\right) \left(\frac{4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + 3 + \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}\right)$$
=
 /         /    /5*pi\\         /    /5*pi\\\ /           /    /5*pi\\         /    /5*pi\\\ 
-|pi + 4*re|asin|----|| + 4*I*im|asin|----|||*|3*pi + 4*re|asin|----|| + 4*I*im|asin|----||| 
 \         \    \ 4  //         \    \ 4  /// \           \    \ 4  //         \    \ 4  /// 
---------------------------------------------------------------------------------------------
                                               2                                             
                                             pi                                              
$$- \frac{\left(\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)} + 4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}\right) \left(4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)} + 3 \pi + 4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}\right)}{\pi^{2}}$$
-(pi + 4*re(asin(5*pi/4)) + 4*i*im(asin(5*pi/4)))*(3*pi + 4*re(asin(5*pi/4)) + 4*i*im(asin(5*pi/4)))/pi^2
Numerical answer [src]
x1 = -3.0 + 2.60301115305176*i
x2 = 5.0 - 2.60301115305176*i
x2 = 5.0 - 2.60301115305176*i