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4sin^2x-sinx-6=0 equation

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Numerical solution:

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The solution

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4*sin (x) - sin(x) - 6 = 0
(4sin⁡2(x)−sin⁡(x))−6=0\left(4 \sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) - 6 = 0
Simplify
Detail solution
Given the equation
(4sin⁡2(x)−sin⁡(x))−6=0\left(4 \sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) - 6 = 0
transform
4sin⁡2(x)−sin⁡(x)−6=04 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} - 6 = 0
(4sin⁡2(x)−sin⁡(x))−6=0\left(4 \sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) - 6 = 0
Do replacement
w=sin⁡(x)w = \sin{\left(x \right)}
This equation is of the form
a*w^2 + b*w + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=4a = 4
b=−1b = -1
c=−6c = -6
, then
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (4) * (-6) = 97

Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

or
w1=18+978w_{1} = \frac{1}{8} + \frac{\sqrt{97}}{8}
w2=18−978w_{2} = \frac{1}{8} - \frac{\sqrt{97}}{8}
do backward replacement
sin⁡(x)=w\sin{\left(x \right)} = w
Given the equation
sin⁡(x)=w\sin{\left(x \right)} = w
- this is the simplest trigonometric equation
This equation is transformed to
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
Or
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, where n - is a integer
substitute w:
x1=2πn+asin⁡(w1)x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
x1=2πn+asin⁡(18+978)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}
x1=2πn+asin⁡(18+978)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(18−978)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}
x2=2πn+asin⁡(18−978)x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn+π−asin⁡(18+978)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}
x3=2πn+π−asin⁡(18+978)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn+π−asin⁡(18−978)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}
x4=2πn+π−asin⁡(18−978)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}
The graph
0-80-60-40-2020406080-1001005-10
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
            /    /      ____\\       /    /      ____\\
            |    |1   \/ 97 ||       |    |1   \/ 97 ||
x1 = pi - re|asin|- - ------|| - I*im|asin|- - ------||
            \    \8     8   //       \    \8     8   //
x1=−re⁡(asin⁡(18−978))+π−iim⁡(asin⁡(18−978))x_{1} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} 
            /    /      ____\\       /    /      ____\\
            |    |1   \/ 97 ||       |    |1   \/ 97 ||
x2 = pi - re|asin|- + ------|| - I*im|asin|- + ------||
            \    \8     8   //       \    \8     8   //
x2=−re⁡(asin⁡(18+978))+π−iim⁡(asin⁡(18+978))x_{2} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} 
         /    /      ____\\     /    /      ____\\
         |    |1   \/ 97 ||     |    |1   \/ 97 ||
x3 = I*im|asin|- - ------|| + re|asin|- - ------||
         \    \8     8   //     \    \8     8   //
x3=re⁡(asin⁡(18−978))+iim⁡(asin⁡(18−978))x_{3} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} 
         /    /      ____\\     /    /      ____\\
         |    |1   \/ 97 ||     |    |1   \/ 97 ||
x4 = I*im|asin|- + ------|| + re|asin|- + ------||
         \    \8     8   //     \    \8     8   //
x4=re⁡(asin⁡(18+978))+iim⁡(asin⁡(18+978))x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} 
x4 = re(asin(1/8 + sqrt(97)/8)) + i*im(asin(1/8 + sqrt(97)/8))
Sum and product of roots [src] 
sum
       /    /      ____\\       /    /      ____\\          /    /      ____\\       /    /      ____\\       /    /      ____\\     /    /      ____\\       /    /      ____\\     /    /      ____\\
       |    |1   \/ 97 ||       |    |1   \/ 97 ||          |    |1   \/ 97 ||       |    |1   \/ 97 ||       |    |1   \/ 97 ||     |    |1   \/ 97 ||       |    |1   \/ 97 ||     |    |1   \/ 97 ||
pi - re|asin|- - ------|| - I*im|asin|- - ------|| + pi - re|asin|- + ------|| - I*im|asin|- + ------|| + I*im|asin|- - ------|| + re|asin|- - ------|| + I*im|asin|- + ------|| + re|asin|- + ------||
       \    \8     8   //       \    \8     8   //          \    \8     8   //       \    \8     8   //       \    \8     8   //     \    \8     8   //       \    \8     8   //     \    \8     8   //
(re⁡(asin⁡(18+978))+iim⁡(asin⁡(18+978)))+(((−re⁡(asin⁡(18−978))+π−iim⁡(asin⁡(18−978)))+(−re⁡(asin⁡(18+978))+π−iim⁡(asin⁡(18+978))))+(re⁡(asin⁡(18−978))+iim⁡(asin⁡(18−978))))\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)}\right) + \left(\left(\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)}\right)\right) 
=
2*pi
2π2 \pi 
product
/       /    /      ____\\       /    /      ____\\\ /       /    /      ____\\       /    /      ____\\\ /    /    /      ____\\     /    /      ____\\\ /    /    /      ____\\     /    /      ____\\\
|       |    |1   \/ 97 ||       |    |1   \/ 97 ||| |       |    |1   \/ 97 ||       |    |1   \/ 97 ||| |    |    |1   \/ 97 ||     |    |1   \/ 97 ||| |    |    |1   \/ 97 ||     |    |1   \/ 97 |||
|pi - re|asin|- - ------|| - I*im|asin|- - ------|||*|pi - re|asin|- + ------|| - I*im|asin|- + ------|||*|I*im|asin|- - ------|| + re|asin|- - ------|||*|I*im|asin|- + ------|| + re|asin|- + ------|||
\       \    \8     8   //       \    \8     8   /// \       \    \8     8   //       \    \8     8   /// \    \    \8     8   //     \    \8     8   /// \    \    \8     8   //     \    \8     8   ///
(−re⁡(asin⁡(18−978))+π−iim⁡(asin⁡(18−978)))(−re⁡(asin⁡(18+978))+π−iim⁡(asin⁡(18+978)))(re⁡(asin⁡(18−978))+iim⁡(asin⁡(18−978)))(re⁡(asin⁡(18+978))+iim⁡(asin⁡(18+978)))\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)}\right) 
=
/    /    /      ____\\     /    /      ____\\\ /    /    /      ____\\     /    /      ____\\\ /          /    /      ____\\     /    /      ____\\\ /          /    /      ____\\     /    /      ____\\\
|    |    |1   \/ 97 ||     |    |1   \/ 97 ||| |    |    |1   \/ 97 ||     |    |1   \/ 97 ||| |          |    |1   \/ 97 ||     |    |1   \/ 97 ||| |          |    |1   \/ 97 ||     |    |1   \/ 97 |||
|I*im|asin|- - ------|| + re|asin|- - ------|||*|I*im|asin|- + ------|| + re|asin|- + ------|||*|-pi + I*im|asin|- - ------|| + re|asin|- - ------|||*|-pi + I*im|asin|- + ------|| + re|asin|- + ------|||
\    \    \8     8   //     \    \8     8   /// \    \    \8     8   //     \    \8     8   /// \          \    \8     8   //     \    \8     8   /// \          \    \8     8   //     \    \8     8   ///
(re⁡(asin⁡(18−978))+iim⁡(asin⁡(18−978)))(re⁡(asin⁡(18+978))+iim⁡(asin⁡(18+978)))(−π+re⁡(asin⁡(18−978))+iim⁡(asin⁡(18−978)))(−π+re⁡(asin⁡(18+978))+iim⁡(asin⁡(18+978)))\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} - \frac{\sqrt{97}}{8} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{8} + \frac{\sqrt{97}}{8} \right)}\right)}\right) 
(i*im(asin(1/8 - sqrt(97)/8)) + re(asin(1/8 - sqrt(97)/8)))*(i*im(asin(1/8 + sqrt(97)/8)) + re(asin(1/8 + sqrt(97)/8)))*(-pi + i*im(asin(1/8 - sqrt(97)/8)) + re(asin(1/8 - sqrt(97)/8)))*(-pi + i*im(asin(1/8 + sqrt(97)/8)) + re(asin(1/8 + sqrt(97)/8)))
Numerical answer [src] 
x1 = 4.71238898038469 - 0.456688335075625*i
x2 = 1.5707963267949 + 0.820700680361328*i
x3 = -1.5707963267949 + 0.456688335075625*i
x4 = 1.5707963267949 - 0.820700680361328*i
x4 = 1.5707963267949 - 0.820700680361328*i
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