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

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Numerical solution:

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The solution

You have entered [src]
     /x   pi\    
2*sin|- - --| = 3
     \3   4 /    
$$2 \sin{\left(\frac{x}{3} - \frac{\pi}{4} \right)} = 3$$
Detail solution
Given the equation
$$2 \sin{\left(\frac{x}{3} - \frac{\pi}{4} \right)} = 3$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -2

The equation is transformed to
$$\cos{\left(\frac{x}{3} + \frac{\pi}{4} \right)} = - \frac{3}{2}$$
As right part of the equation
modulo =
True

but cos
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
The graph
Rapid solution [src]
                        3*pi                     
x1 = 3*re(acos(-3/2)) - ---- + 3*I*im(acos(-3/2))
                         4                       
$$x_{1} = - \frac{3 \pi}{4} + 3 \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + 3 i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}$$
                         21*pi                     
x2 = -3*re(acos(-3/2)) + ----- - 3*I*im(acos(-3/2))
                           4                       
$$x_{2} = - 3 \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + \frac{21 \pi}{4} - 3 i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}$$
x2 = -3*re(acos(-3/2)) + 21*pi/4 - 3*i*im(acos(-3/2))
Sum and product of roots [src]
sum
                   3*pi                                            21*pi                     
3*re(acos(-3/2)) - ---- + 3*I*im(acos(-3/2)) + -3*re(acos(-3/2)) + ----- - 3*I*im(acos(-3/2))
                    4                                                4                       
$$\left(- \frac{3 \pi}{4} + 3 \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + 3 i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}\right) + \left(- 3 \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + \frac{21 \pi}{4} - 3 i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}\right)$$
=
9*pi
----
 2  
$$\frac{9 \pi}{2}$$
product
/                   3*pi                     \ /                    21*pi                     \
|3*re(acos(-3/2)) - ---- + 3*I*im(acos(-3/2))|*|-3*re(acos(-3/2)) + ----- - 3*I*im(acos(-3/2))|
\                    4                       / \                      4                       /
$$\left(- \frac{3 \pi}{4} + 3 \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + 3 i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}\right) \left(- 3 \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + \frac{21 \pi}{4} - 3 i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}\right)$$
=
-9*(-pi + 4*re(acos(-3/2)) + 4*I*im(acos(-3/2)))*(-7*pi + 4*re(acos(-3/2)) + 4*I*im(acos(-3/2)))
------------------------------------------------------------------------------------------------
                                               16                                               
$$- \frac{9 \left(- 7 \pi + 4 \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + 4 i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}\right) \left(- \pi + 4 \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + 4 i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}\right)}{16}$$
-9*(-pi + 4*re(acos(-3/2)) + 4*i*im(acos(-3/2)))*(-7*pi + 4*re(acos(-3/2)) + 4*i*im(acos(-3/2)))/16
Numerical answer [src]
x1 = 7.06858347057703 - 2.88727095035762*i
x2 = 7.06858347057703 + 2.88727095035762*i
x2 = 7.06858347057703 + 2.88727095035762*i