sin(x)*cos(x)=1 la ecuación
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Solución
Gráfica
/ / _____________\\ / / _____________\\
| | ___ ___ / ___ || | | ___ ___ / ___ ||
| | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 || | | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 ||
x1 = 2*re|atan|- - + ------- + ----------------------|| + 2*I*im|atan|- - + ------- + ----------------------||
\ \ 2 2 2 // \ \ 2 2 2 //
x1=2re(atan(−21+221−3i+23i))+2iim(atan(−21+221−3i+23i))
/ / _____________\\ / / _____________\\
| | ___ ___ / ___ || | | ___ ___ / ___ ||
| |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 ||
x2 = - 2*re|atan|- + ------- - ----------------------|| - 2*I*im|atan|- + ------- - ----------------------||
\ \2 2 2 // \ \2 2 2 //
x2=−2re(atan(21−221+3i+23i))−2iim(atan(21−221+3i+23i))
/ / _____________\\ / / _____________\\
| | ___ ___ / ___ || | | ___ ___ / ___ ||
| |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 ||
x3 = - 2*re|atan|- + ------- + ----------------------|| - 2*I*im|atan|- + ------- + ----------------------||
\ \2 2 2 // \ \2 2 2 //
x3=−2re(atan(21+221+3i+23i))−2iim(atan(21+221+3i+23i))
/ / _____________ \\ / / _____________ \\
| | ___ / ___ ___|| | | ___ / ___ ___||
| |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 || | |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 ||
x4 = - 2*re|atan|- + ---------------------- - -------|| - 2*I*im|atan|- + ---------------------- - -------||
\ \2 2 2 // \ \2 2 2 //
x4=−2re(atan(21−23i+221−3i))−2iim(atan(21−23i+221−3i))
x4 = -2*re(atan(1/2 - sqrt(3)*i/2 + sqrt(2)*sqrt(1 - sqrt(3)*i)/2)) - 2*i*im(atan(1/2 - sqrt(3)*i/2 + sqrt(2)*sqrt(1 - sqrt(3)*i)/2))
Suma y producto de raíces
[src]
/ / _____________\\ / / _____________\\ / / _____________\\ / / _____________\\ / / _____________\\ / / _____________\\ / / _____________ \\ / / _____________ \\
| | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ / ___ ___|| | | ___ / ___ ___||
| | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 || | | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 || | |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 ||
2*re|atan|- - + ------- + ----------------------|| + 2*I*im|atan|- - + ------- + ----------------------|| + - 2*re|atan|- + ------- - ----------------------|| - 2*I*im|atan|- + ------- - ----------------------|| + - 2*re|atan|- + ------- + ----------------------|| - 2*I*im|atan|- + ------- + ----------------------|| + - 2*re|atan|- + ---------------------- - -------|| - 2*I*im|atan|- + ---------------------- - -------||
\ \ 2 2 2 // \ \ 2 2 2 // \ \2 2 2 // \ \2 2 2 // \ \2 2 2 // \ \2 2 2 // \ \2 2 2 // \ \2 2 2 //
(((−2re(atan(21−221+3i+23i))−2iim(atan(21−221+3i+23i)))+(2re(atan(−21+221−3i+23i))+2iim(atan(−21+221−3i+23i))))+(−2re(atan(21+221+3i+23i))−2iim(atan(21+221+3i+23i))))+(−2re(atan(21−23i+221−3i))−2iim(atan(21−23i+221−3i)))
/ / _____________\\ / / _____________\\ / / _____________ \\ / / _____________\\ / / _____________\\ / / _____________\\ / / _____________ \\ / / _____________\\
| | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ / ___ ___|| | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ / ___ ___|| | | ___ ___ / ___ ||
| |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 || | | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 || | | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 ||
- 2*re|atan|- + ------- + ----------------------|| - 2*re|atan|- + ------- - ----------------------|| - 2*re|atan|- + ---------------------- - -------|| + 2*re|atan|- - + ------- + ----------------------|| - 2*I*im|atan|- + ------- + ----------------------|| - 2*I*im|atan|- + ------- - ----------------------|| - 2*I*im|atan|- + ---------------------- - -------|| + 2*I*im|atan|- - + ------- + ----------------------||
\ \2 2 2 // \ \2 2 2 // \ \2 2 2 // \ \ 2 2 2 // \ \2 2 2 // \ \2 2 2 // \ \2 2 2 // \ \ 2 2 2 //
−2re(atan(21+221+3i+23i))−2re(atan(21−23i+221−3i))−2re(atan(21−221+3i+23i))+2re(atan(−21+221−3i+23i))−2iim(atan(21+221+3i+23i))−2iim(atan(21−221+3i+23i))−2iim(atan(21−23i+221−3i))+2iim(atan(−21+221−3i+23i))
/ / / _____________\\ / / _____________\\\ / / / _____________\\ / / _____________\\\ / / / _____________\\ / / _____________\\\ / / / _____________ \\ / / _____________ \\\
| | | ___ ___ / ___ || | | ___ ___ / ___ ||| | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | | | ___ / ___ ___|| | | ___ / ___ ___|||
| | | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 || | | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 ||| | | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 ||| | | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 ||| | | |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 || | |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 |||
|2*re|atan|- - + ------- + ----------------------|| + 2*I*im|atan|- - + ------- + ----------------------|||*|- 2*re|atan|- + ------- - ----------------------|| - 2*I*im|atan|- + ------- - ----------------------|||*|- 2*re|atan|- + ------- + ----------------------|| - 2*I*im|atan|- + ------- + ----------------------|||*|- 2*re|atan|- + ---------------------- - -------|| - 2*I*im|atan|- + ---------------------- - -------|||
\ \ \ 2 2 2 // \ \ 2 2 2 /// \ \ \2 2 2 // \ \2 2 2 /// \ \ \2 2 2 // \ \2 2 2 /// \ \ \2 2 2 // \ \2 2 2 ///
(2re(atan(−21+221−3i+23i))+2iim(atan(−21+221−3i+23i)))(−2re(atan(21−221+3i+23i))−2iim(atan(21−221+3i+23i)))(−2re(atan(21+221+3i+23i))−2iim(atan(21+221+3i+23i)))(−2re(atan(21−23i+221−3i))−2iim(atan(21−23i+221−3i)))
/ / / _____________\\ / / _____________\\\ / / / _____________\\ / / _____________\\\ / / / _____________ \\ / / _____________ \\\ / / / _____________\\ / / _____________\\\
| | | ___ ___ / ___ || | | ___ ___ / ___ ||| | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | | | ___ / ___ ___|| | | ___ / ___ ___||| | | | ___ ___ / ___ || | | ___ ___ / ___ |||
| | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 ||| | | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 || | |1 I*\/ 3 \/ 2 *\/ 1 + I*\/ 3 ||| | | |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 || | |1 \/ 2 *\/ 1 - I*\/ 3 I*\/ 3 ||| | | | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 || | | 1 I*\/ 3 \/ 2 *\/ 1 - I*\/ 3 |||
-16*|I*im|atan|- + ------- + ----------------------|| + re|atan|- + ------- + ----------------------|||*|I*im|atan|- + ------- - ----------------------|| + re|atan|- + ------- - ----------------------|||*|I*im|atan|- + ---------------------- - -------|| + re|atan|- + ---------------------- - -------|||*|I*im|atan|- - + ------- + ----------------------|| + re|atan|- - + ------- + ----------------------|||
\ \ \2 2 2 // \ \2 2 2 /// \ \ \2 2 2 // \ \2 2 2 /// \ \ \2 2 2 // \ \2 2 2 /// \ \ \ 2 2 2 // \ \ 2 2 2 ///
−16(re(atan(−21+221−3i+23i))+iim(atan(−21+221−3i+23i)))(re(atan(21−221+3i+23i))+iim(atan(21−221+3i+23i)))(re(atan(21+221+3i+23i))+iim(atan(21+221+3i+23i)))(re(atan(21−23i+221−3i))+iim(atan(21−23i+221−3i)))
-16*(i*im(atan(1/2 + i*sqrt(3)/2 + sqrt(2)*sqrt(1 + i*sqrt(3))/2)) + re(atan(1/2 + i*sqrt(3)/2 + sqrt(2)*sqrt(1 + i*sqrt(3))/2)))*(i*im(atan(1/2 + i*sqrt(3)/2 - sqrt(2)*sqrt(1 + i*sqrt(3))/2)) + re(atan(1/2 + i*sqrt(3)/2 - sqrt(2)*sqrt(1 + i*sqrt(3))/2)))*(i*im(atan(1/2 + sqrt(2)*sqrt(1 - i*sqrt(3))/2 - i*sqrt(3)/2)) + re(atan(1/2 + sqrt(2)*sqrt(1 - i*sqrt(3))/2 - i*sqrt(3)/2)))*(i*im(atan(-1/2 + i*sqrt(3)/2 + sqrt(2)*sqrt(1 - i*sqrt(3))/2)) + re(atan(-1/2 + i*sqrt(3)/2 + sqrt(2)*sqrt(1 - i*sqrt(3))/2)))
x1 = 0.785398163397448 + 0.658478948462408*i
x2 = 0.785398163397448 - 0.658478948462408*i
x3 = -2.35619449019234 - 0.658478948462408*i
x4 = -2.35619449019234 + 0.658478948462408*i
x4 = -2.35619449019234 + 0.658478948462408*i