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(a*sin(x)+b*cos(x))*c=d la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
(a*sin(x) + b*cos(x))*c = d
c(asin⁡(x)+bcos⁡(x))=dc \left(a \sin{\left(x \right)} + b \cos{\left(x \right)}\right) = d
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Gráfica
Respuesta rápida [src] 
         /    /     ______________________      \\         /    /     ______________________      \\
         |    |    /    2    2  2    2  2       ||         |    |    /    2    2  2    2  2       ||
         |    |- \/  - d  + a *c  + b *c   + a*c||         |    |- \/  - d  + a *c  + b *c   + a*c||
x1 = 2*re|atan|---------------------------------|| + 2*I*im|atan|---------------------------------||
         \    \             d + b*c             //         \    \             d + b*c             //
x1=2re⁡(atan⁡(ac−a2c2+b2c2−d2bc+d))+2iim⁡(atan⁡(ac−a2c2+b2c2−d2bc+d))x_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} 
         /    /   ______________________      \\         /    /   ______________________      \\
         |    |  /    2    2  2    2  2       ||         |    |  /    2    2  2    2  2       ||
         |    |\/  - d  + a *c  + b *c   + a*c||         |    |\/  - d  + a *c  + b *c   + a*c||
x2 = 2*re|atan|-------------------------------|| + 2*I*im|atan|-------------------------------||
         \    \            d + b*c            //         \    \            d + b*c            //
x2=2re⁡(atan⁡(ac+a2c2+b2c2−d2bc+d))+2iim⁡(atan⁡(ac+a2c2+b2c2−d2bc+d))x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} 
x2 = 2*re(atan((a*c + sqrt(a^2*c^2 + b^2*c^2 - d^2))/(b*c + d))) + 2*i*im(atan((a*c + sqrt(a^2*c^2 + b^2*c^2 - d^2))/(b*c + d)))
Suma y producto de raíces [src] 
suma
    /    /     ______________________      \\         /    /     ______________________      \\       /    /   ______________________      \\         /    /   ______________________      \\
    |    |    /    2    2  2    2  2       ||         |    |    /    2    2  2    2  2       ||       |    |  /    2    2  2    2  2       ||         |    |  /    2    2  2    2  2       ||
    |    |- \/  - d  + a *c  + b *c   + a*c||         |    |- \/  - d  + a *c  + b *c   + a*c||       |    |\/  - d  + a *c  + b *c   + a*c||         |    |\/  - d  + a *c  + b *c   + a*c||
2*re|atan|---------------------------------|| + 2*I*im|atan|---------------------------------|| + 2*re|atan|-------------------------------|| + 2*I*im|atan|-------------------------------||
    \    \             d + b*c             //         \    \             d + b*c             //       \    \            d + b*c            //         \    \            d + b*c            //
(2re⁡(atan⁡(ac−a2c2+b2c2−d2bc+d))+2iim⁡(atan⁡(ac−a2c2+b2c2−d2bc+d)))+(2re⁡(atan⁡(ac+a2c2+b2c2−d2bc+d))+2iim⁡(atan⁡(ac+a2c2+b2c2−d2bc+d)))\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right) 
=
    /    /   ______________________      \\       /    /     ______________________      \\         /    /   ______________________      \\         /    /     ______________________      \\
    |    |  /    2    2  2    2  2       ||       |    |    /    2    2  2    2  2       ||         |    |  /    2    2  2    2  2       ||         |    |    /    2    2  2    2  2       ||
    |    |\/  - d  + a *c  + b *c   + a*c||       |    |- \/  - d  + a *c  + b *c   + a*c||         |    |\/  - d  + a *c  + b *c   + a*c||         |    |- \/  - d  + a *c  + b *c   + a*c||
2*re|atan|-------------------------------|| + 2*re|atan|---------------------------------|| + 2*I*im|atan|-------------------------------|| + 2*I*im|atan|---------------------------------||
    \    \            d + b*c            //       \    \             d + b*c             //         \    \            d + b*c            //         \    \             d + b*c             //
2re⁡(atan⁡(ac−a2c2+b2c2−d2bc+d))+2re⁡(atan⁡(ac+a2c2+b2c2−d2bc+d))+2iim⁡(atan⁡(ac−a2c2+b2c2−d2bc+d))+2iim⁡(atan⁡(ac+a2c2+b2c2−d2bc+d))2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} 
producto
/    /    /     ______________________      \\         /    /     ______________________      \\\ /    /    /   ______________________      \\         /    /   ______________________      \\\
|    |    |    /    2    2  2    2  2       ||         |    |    /    2    2  2    2  2       ||| |    |    |  /    2    2  2    2  2       ||         |    |  /    2    2  2    2  2       |||
|    |    |- \/  - d  + a *c  + b *c   + a*c||         |    |- \/  - d  + a *c  + b *c   + a*c||| |    |    |\/  - d  + a *c  + b *c   + a*c||         |    |\/  - d  + a *c  + b *c   + a*c|||
|2*re|atan|---------------------------------|| + 2*I*im|atan|---------------------------------|||*|2*re|atan|-------------------------------|| + 2*I*im|atan|-------------------------------|||
\    \    \             d + b*c             //         \    \             d + b*c             /// \    \    \            d + b*c            //         \    \            d + b*c            ///
(2re⁡(atan⁡(ac−a2c2+b2c2−d2bc+d))+2iim⁡(atan⁡(ac−a2c2+b2c2−d2bc+d)))(2re⁡(atan⁡(ac+a2c2+b2c2−d2bc+d))+2iim⁡(atan⁡(ac+a2c2+b2c2−d2bc+d)))\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right) 
=
  /    /    /   ______________________      \\     /    /   ______________________      \\\ /    /    /     ______________________      \\     /    /     ______________________      \\\
  |    |    |  /    2    2  2    2  2       ||     |    |  /    2    2  2    2  2       ||| |    |    |    /    2    2  2    2  2       ||     |    |    /    2    2  2    2  2       |||
  |    |    |\/  - d  + a *c  + b *c   + a*c||     |    |\/  - d  + a *c  + b *c   + a*c||| |    |    |- \/  - d  + a *c  + b *c   + a*c||     |    |- \/  - d  + a *c  + b *c   + a*c|||
4*|I*im|atan|-------------------------------|| + re|atan|-------------------------------|||*|I*im|atan|---------------------------------|| + re|atan|---------------------------------|||
  \    \    \            d + b*c            //     \    \            d + b*c            /// \    \    \             d + b*c             //     \    \             d + b*c             ///
4(re⁡(atan⁡(ac−a2c2+b2c2−d2bc+d))+iim⁡(atan⁡(ac−a2c2+b2c2−d2bc+d)))(re⁡(atan⁡(ac+a2c2+b2c2−d2bc+d))+iim⁡(atan⁡(ac+a2c2+b2c2−d2bc+d)))4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right) 
4*(i*im(atan((sqrt(-d^2 + a^2*c^2 + b^2*c^2) + a*c)/(d + b*c))) + re(atan((sqrt(-d^2 + a^2*c^2 + b^2*c^2) + a*c)/(d + b*c))))*(i*im(atan((-sqrt(-d^2 + a^2*c^2 + b^2*c^2) + a*c)/(d + b*c))) + re(atan((-sqrt(-d^2 + a^2*c^2 + b^2*c^2) + a*c)/(d + b*c))))
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