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sin(x)-cos^2(x)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 sin(x) - cos (x) = 0
$$\sin{\left(x \right)} - \cos^{2}{\left(x \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$\sin{\left(x \right)} - \cos^{2}{\left(x \right)} = 0$$
cambiamos
$$\sin{\left(x \right)} - \cos^{2}{\left(x \right)} = 0$$
$$\sin^{2}{\left(x \right)} + \sin{\left(x \right)} - 1 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 1$$
$$c = -1$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$w_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{5}}{2} - \frac{1}{2} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{\sqrt{5}}{2} - \frac{1}{2} \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$\sin{\left(x \right)} - \cos^{2}{\left(x \right)} = 0$$
cambiamos
$$\sin{\left(x \right)} - \cos^{2}{\left(x \right)} = 0$$
$$\sin^{2}{\left(x \right)} + \sin{\left(x \right)} - 1 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Es la ecuación de la forma
a*w^2 + b*w + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 1$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(1)^2 - 4 * (1) * (-1) = 5
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$w_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{5}}{2} - \frac{1}{2} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{\sqrt{5}}{2} - \frac{1}{2} \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
Respuesta rápida
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/ / ___________\\ / / ___________\\
| | ___ ___ / ___ || | | ___ ___ / ___ ||
| | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||
x1 = - 2*re|atan|- - + ----- + --------------------|| - 2*I*im|atan|- - + ----- + --------------------||
\ \ 2 2 2 // \ \ 2 2 2 //
$$x_{1} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}$$
/ ___________\
| ___ ___ / ___ |
|1 \/ 5 \/ 2 *\/ 1 + \/ 5 |
x2 = 2*atan|- + ----- + --------------------|
\2 2 2 /
$$x_{2} = 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)}$$
/ / ___________\\ / / ___________\\
| | ___ ___ / ___ || | | ___ ___ / ___ ||
| |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||
x3 = 2*re|atan|- - ----- + --------------------|| + 2*I*im|atan|- - ----- + --------------------||
\ \2 2 2 // \ \2 2 2 //
$$x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}$$
/ ___________\
| ___ ___ / ___ |
|1 \/ 5 \/ 2 *\/ 1 + \/ 5 |
x4 = 2*atan|- + ----- - --------------------|
\2 2 2 /
$$x_{4} = 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
x4 = 2*atan(-sqrt(2)*sqrt(1 + sqrt(5))/2 + 1/2 + sqrt(5)/2)
Suma y producto de raíces
[src]
suma
/ / ___________\\ / / ___________\\ / ___________\ / / ___________\\ / / ___________\\ / ___________\
| | ___ ___ / ___ || | | ___ ___ / ___ || | ___ ___ / ___ | | | ___ ___ / ___ || | | ___ ___ / ___ || | ___ ___ / ___ |
| | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || |1 \/ 5 \/ 2 *\/ 1 + \/ 5 |
- 2*re|atan|- - + ----- + --------------------|| - 2*I*im|atan|- - + ----- + --------------------|| + 2*atan|- + ----- + --------------------| + 2*re|atan|- - ----- + --------------------|| + 2*I*im|atan|- - ----- + --------------------|| + 2*atan|- + ----- - --------------------|
\ \ 2 2 2 // \ \ 2 2 2 // \2 2 2 / \ \2 2 2 // \ \2 2 2 // \2 2 2 /
$$\left(\left(2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)} + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right)\right) + 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
=
/ / ___________\\ / ___________\ / ___________\ / / ___________\\ / / ___________\\ / / ___________\\
| | ___ ___ / ___ || | ___ ___ / ___ | | ___ ___ / ___ | | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ ___ / ___ ||
| | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||
- 2*re|atan|- - + ----- + --------------------|| + 2*atan|- + ----- + --------------------| + 2*atan|- + ----- - --------------------| + 2*re|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 //
$$- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}$$
producto
/ / / ___________\\ / / ___________\\\ / ___________\ / / / ___________\\ / / ___________\\\ / ___________\ | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | ___ ___ / ___ | | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | ___ ___ / ___ | | | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||| |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | | | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||| |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | |- 2*re|atan|- - + ----- + --------------------|| - 2*I*im|atan|- - + ----- + --------------------|||*2*atan|- + ----- + --------------------|*|2*re|atan|- - ----- + --------------------|| + 2*I*im|atan|- - ----- + --------------------|||*2*atan|- + ----- - --------------------| \ \ \ 2 2 2 // \ \ 2 2 2 /// \2 2 2 / \ \ \2 2 2 // \ \2 2 2 /// \2 2 2 /
$$\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right) 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)} \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right) 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
=
/ / / ___________\\ / / ___________\\\ / / / ___________\\ / / ___________\\\ / ___________\ / ___________\
| | | ___ ___ / ___ || | | ___ ___ / ___ ||| | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | ___ ___ / ___ | | ___ ___ / ___ |
| | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||| | | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||| |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | |1 \/ 5 \/ 2 *\/ 1 + \/ 5 |
-16*|I*im|atan|- - ----- + --------------------|| + re|atan|- - ----- + --------------------|||*|I*im|atan|- - + ----- + --------------------|| + re|atan|- - + ----- + --------------------|||*atan|- + ----- + --------------------|*atan|- + ----- - --------------------|
\ \ \2 2 2 // \ \2 2 2 /// \ \ \ 2 2 2 // \ \ 2 2 2 /// \2 2 2 / \2 2 2 /
$$- 16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right) \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)} \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
-16*(i*im(atan(1/2 - sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2)) + re(atan(1/2 - sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2)))*(i*im(atan(-1/2 + sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2)) + re(atan(-1/2 + sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2)))*atan(1/2 + sqrt(5)/2 + sqrt(2)*sqrt(1 + sqrt(5))/2)*atan(1/2 + sqrt(5)/2 - sqrt(2)*sqrt(1 + sqrt(5))/2)
Respuesta numérica
[src]
x1 = -66.6396851578782
x2 = 69.781277811468
x3 = -55.8824283321238
x4 = -68.4487989464829
x5 = 88.6308337330067
x6 = 52.740835678534
x7 = -91.7724263865965
x8 = 46.4576503713544
x9 = -5.61694587468707
x10 = -54.073314543519
x11 = -85.4892410794169
x12 = -62.1656136393034
x13 = -99.8647254823809
x14 = -81.0151695608421
x15 = 15.0417238354565
x16 = -11.9001311818667
x17 = 50.9317218899292
x18 = -47.7901292363394
x19 = 2.47535322109728
x20 = -24.4665017962258
x21 = -22.6573880076211
x22 = 77.8735769072523
x23 = 103.006318135971
x24 = 21.324909142636
x25 = -18.1833164890462
x26 = 44.6485365827496
x27 = 32.0821659683904
x28 = -60.3564998506986
x29 = -37.032872410585
x30 = -79.2060557722373
x31 = 27.6080944498156
x32 = -98.0556116937761
x33 = 8.75853852827686
x34 = 63.4980925042884
x35 = -35.2237586219802
x36 = 84.1567622144319
x37 = -87.2983548680217
x38 = 33.8912797569952
x39 = -49.5992430249442
x40 = 71.5903916000727
x41 = -3.80783208608231
x42 = -10.0910173932619
x43 = 38.36535127557
x44 = 25.7989806612109
x45 = 96.7231328287911
x46 = -16.3742027004415
x47 = 82.3476484258271
x48 = 40.1744650641748
x49 = 0.666239432492515
x50 = 19.5157953540313
x51 = 76.0644631186476
x52 = -72.9228704650578
x53 = 90.4399475216115
x54 = 6.9494247396721
x55 = -30.7496871034054
x56 = 94.9140190401863
x57 = -93.5815401752013
x58 = -43.3160577177646
x59 = 65.3072062928931
x60 = -269.51072877623
x61 = -41.5069439291598
x61 = -41.5069439291598