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cos(x)^(2)+2*sin(2x)-3=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 cos (x) + 2*sin(2*x) - 3 = 0
$$\left(2 \sin{\left(2 x \right)} + \cos^{2}{\left(x \right)}\right) - 3 = 0$$
Respuesta rápida
[src]
/ ___\
|20 - 8*\/ 2 |
atan|------------| / / _________________\\
| ___ | | | / ___ ||
\5 - 2*\/ 2 / |log(17) log\\/ 561 - 340*\/ 2 /|
x1 = ------------------ + I*|------- - -------------------------|
2 \ 2 2 /
$$x_{1} = \frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} + i \left(- \frac{\log{\left(\sqrt{561 - 340 \sqrt{2}} \right)}}{2} + \frac{\log{\left(17 \right)}}{2}\right)$$
/ ___\
|20 + 8*\/ 2 |
atan|------------| / / _________________\\
| ___ | | | / ___ ||
\5 + 2*\/ 2 / |log(17) log\\/ 561 + 340*\/ 2 /|
x2 = ------------------ + I*|------- - -------------------------|
2 \ 2 2 /
$$x_{2} = \frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} + i \left(- \frac{\log{\left(\sqrt{340 \sqrt{2} + 561} \right)}}{2} + \frac{\log{\left(17 \right)}}{2}\right)$$
/ / / ___\\\
| | |20 - 8*\/ 2 |||
| |atan|------------|||
| | | ___ |||
| | \5 - 2*\/ 2 /||
/ / _________________\\ |sin|------------------||
|log(17) |4 / ___ || | \ 2 /|
x3 = -pi + I*|------- - log\\/ 561 - 340*\/ 2 /| + atan|-----------------------|
\ 2 / | / / ___\\|
| | |20 - 8*\/ 2 |||
| |atan|------------|||
| | | ___ |||
| | \5 - 2*\/ 2 /||
|cos|------------------||
\ \ 2 //
$$x_{3} = - \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{561 - 340 \sqrt{2}} \right)} + \frac{\log{\left(17 \right)}}{2}\right)$$
/ / / ___\\\
| | |20 + 8*\/ 2 |||
| |atan|------------|||
| | | ___ |||
| | \5 + 2*\/ 2 /||
/ / _________________\\ |sin|------------------||
|log(17) |4 / ___ || | \ 2 /|
x4 = -pi + I*|------- - log\\/ 561 + 340*\/ 2 /| + atan|-----------------------|
\ 2 / | / / ___\\|
| | |20 + 8*\/ 2 |||
| |atan|------------|||
| | | ___ |||
| | \5 + 2*\/ 2 /||
|cos|------------------||
\ \ 2 //
$$x_{4} = - \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{340 \sqrt{2} + 561} \right)} + \frac{\log{\left(17 \right)}}{2}\right)$$
x4 = -pi + atan(sin(atan((8*sqrt(2) + 20)/(2*sqrt(2) + 5))/2)/cos(atan((8*sqrt(2) + 20)/(2*sqrt(2) + 5))/2)) + i*(-log((340*sqrt(2) + 561)^(1/4)) + log(17)/2)
Suma y producto de raíces
[src]
suma
/ / / ___\\\ / / / ___\\\
| | |20 - 8*\/ 2 ||| | | |20 + 8*\/ 2 |||
/ ___\ / ___\ | |atan|------------||| | |atan|------------|||
|20 - 8*\/ 2 | |20 + 8*\/ 2 | | | | ___ ||| | | | ___ |||
atan|------------| / / _________________\\ atan|------------| / / _________________\\ | | \5 - 2*\/ 2 /|| | | \5 + 2*\/ 2 /||
| ___ | | | / ___ || | ___ | | | / ___ || / / _________________\\ |sin|------------------|| / / _________________\\ |sin|------------------||
\5 - 2*\/ 2 / |log(17) log\\/ 561 - 340*\/ 2 /| \5 + 2*\/ 2 / |log(17) log\\/ 561 + 340*\/ 2 /| |log(17) |4 / ___ || | \ 2 /| |log(17) |4 / ___ || | \ 2 /|
------------------ + I*|------- - -------------------------| + ------------------ + I*|------- - -------------------------| + -pi + I*|------- - log\\/ 561 - 340*\/ 2 /| + atan|-----------------------| + -pi + I*|------- - log\\/ 561 + 340*\/ 2 /| + atan|-----------------------|
2 \ 2 2 / 2 \ 2 2 / \ 2 / | / / ___\\| \ 2 / | / / ___\\|
| | |20 - 8*\/ 2 ||| | | |20 + 8*\/ 2 |||
| |atan|------------||| | |atan|------------|||
| | | ___ ||| | | | ___ |||
| | \5 - 2*\/ 2 /|| | | \5 + 2*\/ 2 /||
|cos|------------------|| |cos|------------------||
\ \ 2 // \ \ 2 //
$$\left(- \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{340 \sqrt{2} + 561} \right)} + \frac{\log{\left(17 \right)}}{2}\right)\right) + \left(\left(\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} + i \left(- \frac{\log{\left(\sqrt{340 \sqrt{2} + 561} \right)}}{2} + \frac{\log{\left(17 \right)}}{2}\right)\right) + \left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} + i \left(- \frac{\log{\left(\sqrt{561 - 340 \sqrt{2}} \right)}}{2} + \frac{\log{\left(17 \right)}}{2}\right)\right)\right) + \left(- \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{561 - 340 \sqrt{2}} \right)} + \frac{\log{\left(17 \right)}}{2}\right)\right)\right)$$
=
/ / / ___\\\ / / / ___\\\
| | |20 - 8*\/ 2 ||| | | |20 + 8*\/ 2 |||
/ ___\ / ___\ | |atan|------------||| | |atan|------------|||
|20 - 8*\/ 2 | |20 + 8*\/ 2 | | | | ___ ||| | | | ___ |||
atan|------------| atan|------------| / / _________________\\ / / _________________\\ | | \5 - 2*\/ 2 /|| | | \5 + 2*\/ 2 /||
| ___ | | ___ | / / _________________\\ / / _________________\\ | | / ___ || | | / ___ || |sin|------------------|| |sin|------------------||
\5 - 2*\/ 2 / \5 + 2*\/ 2 / |log(17) |4 / ___ || |log(17) |4 / ___ || |log(17) log\\/ 561 - 340*\/ 2 /| |log(17) log\\/ 561 + 340*\/ 2 /| | \ 2 /| | \ 2 /|
------------------ + ------------------ - 2*pi + I*|------- - log\\/ 561 - 340*\/ 2 /| + I*|------- - log\\/ 561 + 340*\/ 2 /| + I*|------- - -------------------------| + I*|------- - -------------------------| + atan|-----------------------| + atan|-----------------------|
2 2 \ 2 / \ 2 / \ 2 2 / \ 2 2 / | / / ___\\| | / / ___\\|
| | |20 - 8*\/ 2 ||| | | |20 + 8*\/ 2 |||
| |atan|------------||| | |atan|------------|||
| | | ___ ||| | | | ___ |||
| | \5 - 2*\/ 2 /|| | | \5 + 2*\/ 2 /||
|cos|------------------|| |cos|------------------||
\ \ 2 // \ \ 2 //
$$- 2 \pi + \frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} \right)}} \right)} + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{340 \sqrt{2} + 561} \right)} + \frac{\log{\left(17 \right)}}{2}\right) + i \left(- \frac{\log{\left(\sqrt{340 \sqrt{2} + 561} \right)}}{2} + \frac{\log{\left(17 \right)}}{2}\right) + i \left(- \log{\left(\sqrt[4]{561 - 340 \sqrt{2}} \right)} + \frac{\log{\left(17 \right)}}{2}\right) + i \left(- \frac{\log{\left(\sqrt{561 - 340 \sqrt{2}} \right)}}{2} + \frac{\log{\left(17 \right)}}{2}\right)$$
producto
/ / / / ___\\\\ / / / / ___\\\\
| | | |20 - 8*\/ 2 |||| | | | |20 + 8*\/ 2 ||||
/ / ___\ \ / / ___\ \ | | |atan|------------|||| | | |atan|------------||||
| |20 - 8*\/ 2 | | | |20 + 8*\/ 2 | | | | | | ___ |||| | | | | ___ ||||
|atan|------------| / / _________________\\| |atan|------------| / / _________________\\| | | | \5 - 2*\/ 2 /||| | | | \5 + 2*\/ 2 /|||
| | ___ | | | / ___ ||| | | ___ | | | / ___ ||| | / / _________________\\ |sin|------------------||| | / / _________________\\ |sin|------------------|||
| \5 - 2*\/ 2 / |log(17) log\\/ 561 - 340*\/ 2 /|| | \5 + 2*\/ 2 / |log(17) log\\/ 561 + 340*\/ 2 /|| | |log(17) |4 / ___ || | \ 2 /|| | |log(17) |4 / ___ || | \ 2 /||
|------------------ + I*|------- - -------------------------||*|------------------ + I*|------- - -------------------------||*|-pi + I*|------- - log\\/ 561 - 340*\/ 2 /| + atan|-----------------------||*|-pi + I*|------- - log\\/ 561 + 340*\/ 2 /| + atan|-----------------------||
\ 2 \ 2 2 // \ 2 \ 2 2 // | \ 2 / | / / ___\\|| | \ 2 / | / / ___\\||
| | | |20 - 8*\/ 2 |||| | | | |20 + 8*\/ 2 ||||
| | |atan|------------|||| | | |atan|------------||||
| | | | ___ |||| | | | | ___ ||||
| | | \5 - 2*\/ 2 /||| | | | \5 + 2*\/ 2 /|||
| |cos|------------------||| | |cos|------------------|||
\ \ \ 2 /// \ \ \ 2 ///
$$\left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} + i \left(- \frac{\log{\left(\sqrt{561 - 340 \sqrt{2}} \right)}}{2} + \frac{\log{\left(17 \right)}}{2}\right)\right) \left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} + i \left(- \frac{\log{\left(\sqrt{340 \sqrt{2} + 561} \right)}}{2} + \frac{\log{\left(17 \right)}}{2}\right)\right) \left(- \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{20 - 8 \sqrt{2}}{5 - 2 \sqrt{2}} \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{561 - 340 \sqrt{2}} \right)} + \frac{\log{\left(17 \right)}}{2}\right)\right) \left(- \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{8 \sqrt{2} + 20}{2 \sqrt{2} + 5} \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{340 \sqrt{2} + 561} \right)} + \frac{\log{\left(17 \right)}}{2}\right)\right)$$
=
/ / / ___\ \ \ / / / ___\ \ \ / / / ___\ \ \ / / / ___\ \ \
|I*\- log\33 - 20*\/ 2 / + log(17)/ | |I*\- log\33 + 20*\/ 2 / + log(17)/ | | I*\- log\33 - 20*\/ 2 / + log(17)/ | | I*\- log\33 + 20*\/ 2 / + log(17)/ |
|---------------------------------- + atan(4)|*|---------------------------------- + atan(4)|*|-2*pi + ---------------------------------- + atan(4)|*|-2*pi + ---------------------------------- + atan(4)|
\ 2 / \ 2 / \ 2 / \ 2 /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
16
$$\frac{\left(\operatorname{atan}{\left(4 \right)} + \frac{i \left(- \log{\left(33 - 20 \sqrt{2} \right)} + \log{\left(17 \right)}\right)}{2}\right) \left(\operatorname{atan}{\left(4 \right)} + \frac{i \left(- \log{\left(20 \sqrt{2} + 33 \right)} + \log{\left(17 \right)}\right)}{2}\right) \left(- 2 \pi + \operatorname{atan}{\left(4 \right)} + \frac{i \left(- \log{\left(33 - 20 \sqrt{2} \right)} + \log{\left(17 \right)}\right)}{2}\right) \left(- 2 \pi + \operatorname{atan}{\left(4 \right)} + \frac{i \left(- \log{\left(20 \sqrt{2} + 33 \right)} + \log{\left(17 \right)}\right)}{2}\right)}{16}$$
(i*(-log(33 - 20*sqrt(2)) + log(17))/2 + atan(4))*(i*(-log(33 + 20*sqrt(2)) + log(17))/2 + atan(4))*(-2*pi + i*(-log(33 - 20*sqrt(2)) + log(17))/2 + atan(4))*(-2*pi + i*(-log(33 + 20*sqrt(2)) + log(17))/2 + atan(4))/16
Respuesta numérica
[src]
x1 = -2.47868382175578 + 0.320577469865141*i
x2 = -2.47868382175578 - 0.320577469865141*i
x3 = 0.662908831834016 + 0.320577469865141*i
x4 = 0.662908831834016 - 0.320577469865141*i
x4 = 0.662908831834016 - 0.320577469865141*i