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2*cos(x+pi/3)=5 equation

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Numerical solution:

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The solution

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     /    pi\    
2*cos|x + --| = 5
     \    3 /    
$$2 \cos{\left(x + \frac{\pi}{3} \right)} = 5$$
Detail solution
Given the equation
$$2 \cos{\left(x + \frac{\pi}{3} \right)} = 5$$
- this is the simplest trigonometric equation
Divide both parts of the equation by 2

The equation is transformed to
$$\cos{\left(x + \frac{\pi}{3} \right)} = \frac{5}{2}$$
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]
       pi                                  
x1 = - -- + I*im(acos(5/2)) + re(acos(5/2))
       3                                   
$$x_{1} = - \frac{\pi}{3} + \operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}$$
     5*pi                  
x2 = ---- - I*im(acos(5/2))
      3                    
$$x_{2} = \frac{5 \pi}{3} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}$$
x2 = 5*pi/3 - i*im(acos(5/2))
Sum and product of roots [src]
sum
  pi                                     5*pi                  
- -- + I*im(acos(5/2)) + re(acos(5/2)) + ---- - I*im(acos(5/2))
  3                                       3                    
$$\left(\frac{5 \pi}{3} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right) + \left(- \frac{\pi}{3} + \operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right)$$
=
4*pi                
---- + re(acos(5/2))
 3                  
$$\operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + \frac{4 \pi}{3}$$
product
/  pi                                  \ /5*pi                  \
|- -- + I*im(acos(5/2)) + re(acos(5/2))|*|---- - I*im(acos(5/2))|
\  3                                   / \ 3                    /
$$\left(\frac{5 \pi}{3} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right) \left(- \frac{\pi}{3} + \operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right)$$
=
(5*pi - 3*I*im(acos(5/2)))*(-pi + 3*re(acos(5/2)) + 3*I*im(acos(5/2)))
----------------------------------------------------------------------
                                  9                                   
$$\frac{\left(5 \pi - 3 i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right) \left(- \pi + 3 \operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + 3 i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right)}{9}$$
(5*pi - 3*i*im(acos(5/2)))*(-pi + 3*re(acos(5/2)) + 3*i*im(acos(5/2)))/9
Numerical answer [src]
x1 = -1.0471975511966 + 1.56679923697241*i
x2 = 5.23598775598299 - 1.56679923697241*i
x2 = 5.23598775598299 - 1.56679923697241*i