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Expresiones idénticas
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Expresiones semejantes
- 3tg^3x+3tg^2x+3tgx-1=0
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3tg^3x+3tg^2x+3tgx+1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 3*tan (x) + 3*tan (x) + 3*tan(x) + 1 = 0
$$\left(\left(3 \tan^{3}{\left(x \right)} + 3 \tan^{2}{\left(x \right)}\right) + 3 \tan{\left(x \right)}\right) + 1 = 0$$
Suma y producto de raíces
[src]
suma
/ / / ___\ \\ / / / ___\ \\ / / / ___\\\ / / / ___\\\
| | 2/3 | 1 I*\/ 3 | || | | 2/3 | 1 I*\/ 3 | || | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 |||
| | 2 *|- - - -------| 3 ___ || | | 2 *|- - - -------| 3 ___ || | | 3 ___ 2 *|- - + -------||| | | 3 ___ 2 *|- - + -------||| / 3 ___ 2/3\
| | 1 \ 2 2 / \/ 2 || | | 1 \ 2 2 / \/ 2 || | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /|| |1 \/ 2 2 |
I*im|atan|- - - -------------------- + -----------------|| + re|atan|- - - -------------------- + -----------------|| + - re|atan|- - ----------------- + --------------------|| - I*im|atan|- - ----------------- + --------------------|| - atan|- - ----- + ----|
| | 3 3 / ___\|| | | 3 3 / ___\|| | |3 / ___\ 3 || | |3 / ___\ 3 || \3 3 3 /
| | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | ||
| | 3*|- - - -------||| | | 3*|- - - -------||| | | 3*|- - + -------| || | | 3*|- - + -------| ||
\ \ \ 2 2 /// \ \ \ 2 2 /// \ \ \ 2 2 / // \ \ \ 2 2 / //
$$\left(\left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right) + \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{3} + \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{3} + \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right)\right) - \operatorname{atan}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)}$$
=
/ / / ___\\\ / / / ___\ \\ / / / ___\\\ / / / ___\ \\
| | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 | || | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 | ||
/ 3 ___ 2/3\ | | 3 ___ 2 *|- - + -------||| | | 2 *|- - - -------| 3 ___ || | | 3 ___ 2 *|- - + -------||| | | 2 *|- - - -------| 3 ___ ||
|1 \/ 2 2 | | |1 \/ 2 \ 2 2 /|| | | 1 \ 2 2 / \/ 2 || | |1 \/ 2 \ 2 2 /|| | | 1 \ 2 2 / \/ 2 ||
- atan|- - ----- + ----| - re|atan|- - ----------------- + --------------------|| + I*im|atan|- - - -------------------- + -----------------|| - I*im|atan|- - ----------------- + --------------------|| + re|atan|- - - -------------------- + -----------------||
\3 3 3 / | |3 / ___\ 3 || | | 3 3 / ___\|| | |3 / ___\ 3 || | | 3 3 / ___\||
| | | 1 I*\/ 3 | || | | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 |||
| | 3*|- - + -------| || | | 3*|- - - -------||| | | 3*|- - + -------| || | | 3*|- - - -------|||
\ \ \ 2 2 / // \ \ \ 2 2 /// \ \ \ 2 2 / // \ \ \ 2 2 ///
$$- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{3} + \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} - \operatorname{atan}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{3} + \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}$$
producto
/ / / / ___\ \\ / / / ___\ \\\ / / / / ___\\\ / / / ___\\\\ | | | 2/3 | 1 I*\/ 3 | || | | 2/3 | 1 I*\/ 3 | ||| | | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 |||| | | | 2 *|- - - -------| 3 ___ || | | 2 *|- - - -------| 3 ___ ||| | | | 3 ___ 2 *|- - + -------||| | | 3 ___ 2 *|- - + -------|||| / / 3 ___ 2/3\\ | | | 1 \ 2 2 / \/ 2 || | | 1 \ 2 2 / \/ 2 ||| | | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /||| | |1 \/ 2 2 || |I*im|atan|- - - -------------------- + -----------------|| + re|atan|- - - -------------------- + -----------------|||*|- re|atan|- - ----------------- + --------------------|| - I*im|atan|- - ----------------- + --------------------|||*|-atan|- - ----- + ----|| | | | 3 3 / ___\|| | | 3 3 / ___\||| | | |3 / ___\ 3 || | |3 / ___\ 3 ||| \ \3 3 3 // | | | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 |||| | | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | ||| | | | 3*|- - - -------||| | | 3*|- - - -------|||| | | | 3*|- - + -------| || | | 3*|- - + -------| ||| \ \ \ \ 2 2 /// \ \ \ 2 2 //// \ \ \ \ 2 2 / // \ \ \ 2 2 / ///
$$\left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{3} + \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{3} + \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right) \left(- \operatorname{atan}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)}\right)$$
=
/ / / 3 ___ 2/3 3 ___ ___ 2/3 ___\\ / / 3 ___ 2/3 3 ___ ___ 2/3 ___\\\ / / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\ / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\\ / 3 ___ 2/3\ | | | 1 \/ 2 2 I*\/ 2 *\/ 3 I*2 *\/ 3 || | | 1 \/ 2 2 I*\/ 2 *\/ 3 I*2 *\/ 3 ||| | | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 || | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 ||| |1 \/ 2 2 | |I*im|atan|- - - ----- + ---- + ------------- + ------------|| + re|atan|- - - ----- + ---- + ------------- + ------------|||*|I*im|atan|- - ---- + ----- + ------------- + ------------|| + re|atan|- - ---- + ----- + ------------- + ------------|||*atan|- - ----- + ----| \ \ \ 3 6 6 6 6 // \ \ 3 6 6 6 6 /// \ \ \3 6 6 6 6 // \ \3 6 6 6 6 /// \3 3 3 /
$$\left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{3} - \frac{\sqrt[3]{2}}{6} + \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{3} - \frac{\sqrt[3]{2}}{6} + \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)}\right) \operatorname{atan}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)}$$
(i*im(atan(-1/3 - 2^(1/3)/6 + 2^(2/3)/6 + i*2^(1/3)*sqrt(3)/6 + i*2^(2/3)*sqrt(3)/6)) + re(atan(-1/3 - 2^(1/3)/6 + 2^(2/3)/6 + i*2^(1/3)*sqrt(3)/6 + i*2^(2/3)*sqrt(3)/6)))*(i*im(atan(1/3 - 2^(2/3)/6 + 2^(1/3)/6 + i*2^(1/3)*sqrt(3)/6 + i*2^(2/3)*sqrt(3)/6)) + re(atan(1/3 - 2^(2/3)/6 + 2^(1/3)/6 + i*2^(1/3)*sqrt(3)/6 + i*2^(2/3)*sqrt(3)/6)))*atan(1/3 - 2^(1/3)/3 + 2^(2/3)/3)
Respuesta rápida
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/ / / ___\ \\ / / / ___\ \\
| | 2/3 | 1 I*\/ 3 | || | | 2/3 | 1 I*\/ 3 | ||
| | 2 *|- - - -------| 3 ___ || | | 2 *|- - - -------| 3 ___ ||
| | 1 \ 2 2 / \/ 2 || | | 1 \ 2 2 / \/ 2 ||
x1 = I*im|atan|- - - -------------------- + -----------------|| + re|atan|- - - -------------------- + -----------------||
| | 3 3 / ___\|| | | 3 3 / ___\||
| | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 |||
| | 3*|- - - -------||| | | 3*|- - - -------|||
\ \ \ 2 2 /// \ \ \ 2 2 ///
$$x_{1} = \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{3} + \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{3} + \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}$$
/ / / ___\\\ / / / ___\\\
| | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 |||
| | 3 ___ 2 *|- - + -------||| | | 3 ___ 2 *|- - + -------|||
| |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /||
x2 = - re|atan|- - ----------------- + --------------------|| - I*im|atan|- - ----------------- + --------------------||
| |3 / ___\ 3 || | |3 / ___\ 3 ||
| | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | ||
| | 3*|- - + -------| || | | 3*|- - + -------| ||
\ \ \ 2 2 / // \ \ \ 2 2 / //
$$x_{2} = - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}$$
/ 3 ___ 2/3\
|1 \/ 2 2 |
x3 = -atan|- - ----- + ----|
\3 3 3 /
$$x_{3} = - \operatorname{atan}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)}$$
x3 = -atan(-2^(1/3)/3 + 1/3 + 2^(2/3)/3)
Respuesta numérica
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x1 = -50.6820763300341
x2 = 15.2913693953515
x3 = 34.1409253168903
x4 = 62.4152591991984
x5 = 40.4241106240699
x6 = -3.55818652618723
x7 = -44.3988910228545
x8 = 100.114371042276
x9 = -12.9829644869566
x10 = -82.0980028659321
x11 = -38.115705715675
x12 = -22.407742447726
x13 = 87.5480004279168
x14 = -69.5316322515729
x15 = -100.947558787471
x16 = 78.1232224671474
x17 = -28.6909277549056
x18 = -85.2395955195219
x19 = -16.1245571405464
x20 = -56.9652616372137
x21 = -78.9564102123423
x22 = -31.8325204084954
x23 = -91.5227808267014
x24 = 96.9727783886862
x25 = -47.5404836764443
x26 = -63.2484469443933
x27 = -60.1068542908035
x28 = 74.9816298135576
x29 = 27.8577400097107
x30 = 71.8400371599678
x31 = 24.7161473561209
x32 = 59.2736665456086
x33 = -0.416593872597435
x34 = 81.2648151207372
x35 = -94.6643734802912
x36 = -72.6732249051627
x37 = 68.698444506378
x38 = -88.3811881731116
x39 = 90.6895930815066
x40 = 30.9993326633005
x41 = -53.8236689836239
x42 = 5.86659143458215
x43 = 93.8311857350964
x44 = 52.990481238429
x45 = 46.7072959312495
x46 = 37.2825179704801
x47 = 56.1320738920188
x48 = 18.4329620489413
x49 = -97.805966133881
x50 = -34.9741130620852
x51 = -6.69977917977702
x52 = 84.406407774327
x53 = 2.72499878099236
x54 = 49.8488885848393
x55 = -25.5493351013158
x56 = -75.8148175587525
x57 = -19.2661497941362
x58 = 65.5568518527882
x59 = 9.00818408817194
x60 = -41.2572983692647
x61 = 21.5745547025311
x62 = -66.3900395979831
x63 = -9.84137183336681
x64 = 12.1497767417617
x65 = 43.5657032776597
x65 = 43.5657032776597