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tan(-22.3°)=2*tan(x°) equation

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Numerical solution:

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The solution

You have entered [src]
   //-223*pi\\              
   ||-------||              
   |\   10  /|        /x*pi\
tan|---------| = 2*tan|----|
   \   360   /        \360 /
$$\tan{\left(\frac{\left(-1\right) \frac{223}{10} \pi}{360} \right)} = 2 \tan{\left(\frac{\pi x}{360} \right)}$$
Detail solution
Given the equation
$$\tan{\left(\frac{\left(-1\right) \frac{223}{10} \pi}{360} \right)} = 2 \tan{\left(\frac{\pi x}{360} \right)}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -2

The equation is transformed to
$$\tan{\left(\frac{\pi x}{360} \right)} = - \frac{\tan{\left(\frac{223 \pi}{3600} \right)}}{2}$$
This equation is transformed to
$$\frac{\pi x}{360} = \pi n + \operatorname{atan}{\left(- \frac{\tan{\left(\frac{223 \pi}{3600} \right)}}{2} \right)}$$
Or
$$\frac{\pi x}{360} = \pi n - \operatorname{atan}{\left(\frac{\tan{\left(\frac{223 \pi}{3600} \right)}}{2} \right)}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{\pi}{360}$$
we get the answer:
$$x_{1} = \frac{360 \left(\pi n - \operatorname{atan}{\left(\frac{\tan{\left(\frac{223 \pi}{3600} \right)}}{2} \right)}\right)}{\pi}$$
The graph
Rapid solution [src]
              /   /223*pi\\
              |tan|------||
              |   \ 3600 /|
     -360*atan|-----------|
              \     2     /
x1 = ----------------------
               pi          
$$x_{1} = - \frac{360 \operatorname{atan}{\left(\frac{\tan{\left(\frac{223 \pi}{3600} \right)}}{2} \right)}}{\pi}$$
x1 = -360*atan(tan(223*pi/3600)/2)/pi
Sum and product of roots [src]
sum
         /   /223*pi\\
         |tan|------||
         |   \ 3600 /|
-360*atan|-----------|
         \     2     /
----------------------
          pi          
$$- \frac{360 \operatorname{atan}{\left(\frac{\tan{\left(\frac{223 \pi}{3600} \right)}}{2} \right)}}{\pi}$$
=
         /   /223*pi\\
         |tan|------||
         |   \ 3600 /|
-360*atan|-----------|
         \     2     /
----------------------
          pi          
$$- \frac{360 \operatorname{atan}{\left(\frac{\tan{\left(\frac{223 \pi}{3600} \right)}}{2} \right)}}{\pi}$$
product
         /   /223*pi\\
         |tan|------||
         |   \ 3600 /|
-360*atan|-----------|
         \     2     /
----------------------
          pi          
$$- \frac{360 \operatorname{atan}{\left(\frac{\tan{\left(\frac{223 \pi}{3600} \right)}}{2} \right)}}{\pi}$$
=
         /   /223*pi\\
         |tan|------||
         |   \ 3600 /|
-360*atan|-----------|
         \     2     /
----------------------
          pi          
$$- \frac{360 \operatorname{atan}{\left(\frac{\tan{\left(\frac{223 \pi}{3600} \right)}}{2} \right)}}{\pi}$$
-360*atan(tan(223*pi/3600)/2)/pi
Numerical answer [src]
x1 = -11.2565712594489
x1 = -11.2565712594489