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Expresiones idénticas
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(a*sin(x)+b*cos(x))*c=d la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
(a*sin(x) + b*cos(x))*c = d
$$c \left(a \sin{\left(x \right)} + b \cos{\left(x \right)}\right) = d$$
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 //
$$x_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}$$
/ / ______________________ \\ / / ______________________ \\
| | / 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 //
$$x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}$$
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 //
$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right)$$
=
/ / ______________________ \\ / / ______________________ \\ / / ______________________ \\ / / ______________________ \\
| | / 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 //
$$2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}$$
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 ///
$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right)$$
=
/ / / ______________________ \\ / / ______________________ \\\ / / / ______________________ \\ / / ______________________ \\\ | | | / 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 ///
$$4 \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c - \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{a c + \sqrt{a^{2} c^{2} + b^{2} c^{2} - d^{2}}}{b c + d} \right)}\right)}\right)$$
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))))