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cos^2(x)+3*sin(x)+1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 cos (x) + 3*sin(x) + 1 = 0
$$\left(3 \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(3 \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + 1 = 0$$
cambiamos
$$3 \sin{\left(x \right)} + \cos^{2}{\left(x \right)} + 1 = 0$$
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} + 2 = 0$$
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
$$a = -1$$
$$b = 3$$
$$c = 2$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{3}{2} - \frac{\sqrt{17}}{2}$$
$$w_{2} = \frac{3}{2} + \frac{\sqrt{17}}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} - \frac{\sqrt{17}}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} - \frac{\sqrt{17}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} + \frac{\sqrt{17}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} + \frac{\sqrt{17}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{3}{2} - \frac{\sqrt{17}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{3}{2} - \frac{\sqrt{17}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} + \frac{\sqrt{17}}{2} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} + \frac{\sqrt{17}}{2} \right)}$$
$$\left(3 \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + 1 = 0$$
cambiamos
$$3 \sin{\left(x \right)} + \cos^{2}{\left(x \right)} + 1 = 0$$
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} + 2 = 0$$
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
$$a = -1$$
$$b = 3$$
$$c = 2$$
, entonces
D = b^2 - 4 * a * c =
(3)^2 - 4 * (-1) * (2) = 17
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{3}{2} - \frac{\sqrt{17}}{2}$$
$$w_{2} = \frac{3}{2} + \frac{\sqrt{17}}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} - \frac{\sqrt{17}}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} - \frac{\sqrt{17}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} + \frac{\sqrt{17}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} + \frac{\sqrt{17}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{3}{2} - \frac{\sqrt{17}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{3}{2} - \frac{\sqrt{17}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} + \frac{\sqrt{17}}{2} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} + \frac{\sqrt{17}}{2} \right)}$$
Respuesta rápida
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/ / ______________\\ / / ______________\\
| | ____ ___ / ____ || | | ____ ___ / ____ ||
| | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 ||
x1 = 2*re|atan|- - + ------ + -----------------------|| + 2*I*im|atan|- - + ------ + -----------------------||
\ \ 4 4 4 // \ \ 4 4 4 //
$$x_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)}$$
/ ______________\
| ____ ___ / ____ |
|3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 |
x2 = -2*atan|- + ------ + -----------------------|
\4 4 4 /
$$x_{2} = - 2 \operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} \right)}$$
/ / ______________\\ / / ______________\\
| | ____ ___ / ____ || | | ____ ___ / ____ ||
| |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 ||
x3 = - 2*re|atan|- - ------ + -----------------------|| - 2*I*im|atan|- - ------ + -----------------------||
\ \4 4 4 // \ \4 4 4 //
$$x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)}$$
/ ______________\
| ____ ___ / ____ |
|3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 |
x4 = -2*atan|- + ------ - -----------------------|
\4 4 4 /
$$x_{4} = - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} + \frac{3}{4} + \frac{\sqrt{17}}{4} \right)}$$
x4 = -2*atan(-sqrt(2)*sqrt(5 + 3*sqrt(17))/4 + 3/4 + sqrt(17)/4)
Suma y producto de raíces
[src]
suma
/ / ______________\\ / / ______________\\ / ______________\ / / ______________\\ / / ______________\\ / ______________\
| | ____ ___ / ____ || | | ____ ___ / ____ || | ____ ___ / ____ | | | ____ ___ / ____ || | | ____ ___ / ____ || | ____ ___ / ____ |
| | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || |3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 | | |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || |3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 |
2*re|atan|- - + ------ + -----------------------|| + 2*I*im|atan|- - + ------ + -----------------------|| - 2*atan|- + ------ + -----------------------| + - 2*re|atan|- - ------ + -----------------------|| - 2*I*im|atan|- - ------ + -----------------------|| - 2*atan|- + ------ - -----------------------|
\ \ 4 4 4 // \ \ 4 4 4 // \4 4 4 / \ \4 4 4 // \ \4 4 4 // \4 4 4 /
$$- 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} + \frac{3}{4} + \frac{\sqrt{17}}{4} \right)} + \left(\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)}\right) + \left(- 2 \operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} \right)} + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)}\right)\right)\right)$$
=
/ ______________\ / ______________\ / / ______________\\ / / ______________\\ / / ______________\\ / / ______________\\
| ____ ___ / ____ | | ____ ___ / ____ | | | ____ ___ / ____ || | | ____ ___ / ____ || | | ____ ___ / ____ || | | ____ ___ / ____ ||
|3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 | |3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 | | |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 ||
- 2*atan|- + ------ - -----------------------| - 2*atan|- + ------ + -----------------------| - 2*re|atan|- - ------ + -----------------------|| + 2*re|atan|- - + ------ + -----------------------|| - 2*I*im|atan|- - ------ + -----------------------|| + 2*I*im|atan|- - + ------ + -----------------------||
\4 4 4 / \4 4 4 / \ \4 4 4 // \ \ 4 4 4 // \ \4 4 4 // \ \ 4 4 4 //
$$- 2 \operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} \right)} - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} + \frac{3}{4} + \frac{\sqrt{17}}{4} \right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)}$$
producto
/ / / ______________\\ / / ______________\\\ / ______________\ / / / ______________\\ / / ______________\\\ / ______________\ | | | ____ ___ / ____ || | | ____ ___ / ____ ||| | ____ ___ / ____ | | | | ____ ___ / ____ || | | ____ ___ / ____ ||| | ____ ___ / ____ | | | | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 ||| |3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 | | | |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 ||| |3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 | |2*re|atan|- - + ------ + -----------------------|| + 2*I*im|atan|- - + ------ + -----------------------|||*-2*atan|- + ------ + -----------------------|*|- 2*re|atan|- - ------ + -----------------------|| - 2*I*im|atan|- - ------ + -----------------------|||*-2*atan|- + ------ - -----------------------| \ \ \ 4 4 4 // \ \ 4 4 4 /// \4 4 4 / \ \ \4 4 4 // \ \4 4 4 /// \4 4 4 /
$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} \right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} + \frac{3}{4} + \frac{\sqrt{17}}{4} \right)}\right)$$
=
/ / / ______________\\ / / ______________\\\ / / / ______________\\ / / ______________\\\ / ______________\ / ______________\
| | | ____ ___ / ____ || | | ____ ___ / ____ ||| | | | ____ ___ / ____ || | | ____ ___ / ____ ||| | ____ ___ / ____ | | ____ ___ / ____ |
| | | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | | 3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 ||| | | |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 || | |3 \/ 17 \/ 2 *\/ 5 - 3*\/ 17 ||| |3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 | |3 \/ 17 \/ 2 *\/ 5 + 3*\/ 17 |
-16*|I*im|atan|- - + ------ + -----------------------|| + re|atan|- - + ------ + -----------------------|||*|I*im|atan|- - ------ + -----------------------|| + re|atan|- - ------ + -----------------------|||*atan|- + ------ - -----------------------|*atan|- + ------ + -----------------------|
\ \ \ 4 4 4 // \ \ 4 4 4 /// \ \ \4 4 4 // \ \4 4 4 /// \4 4 4 / \4 4 4 /
$$- 16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{3}{4} + \frac{\sqrt{2} \sqrt{5 - 3 \sqrt{17}}}{4} \right)}\right)}\right) \operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} \right)} \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{5 + 3 \sqrt{17}}}{4} + \frac{3}{4} + \frac{\sqrt{17}}{4} \right)}$$
-16*(i*im(atan(-3/4 + sqrt(17)/4 + sqrt(2)*sqrt(5 - 3*sqrt(17))/4)) + re(atan(-3/4 + sqrt(17)/4 + sqrt(2)*sqrt(5 - 3*sqrt(17))/4)))*(i*im(atan(3/4 - sqrt(17)/4 + sqrt(2)*sqrt(5 - 3*sqrt(17))/4)) + re(atan(3/4 - sqrt(17)/4 + sqrt(2)*sqrt(5 - 3*sqrt(17))/4)))*atan(3/4 + sqrt(17)/4 - sqrt(2)*sqrt(5 + 3*sqrt(17))/4)*atan(3/4 + sqrt(17)/4 + sqrt(2)*sqrt(5 + 3*sqrt(17))/4)
Respuesta numérica
[src]
x1 = -71.6603697800178
x2 = -138.826338010498
x3 = -88.5608555530617
x4 = 43.3860358977096
x5 = -101.127226167421
x6 = 68.518777126428
x7 = -21.3948873225811
x8 = 49.6692212048892
x9 = -107.4104114746
x10 = 54.003336363574
x11 = 47.7201510563944
x12 = -69.7112996315229
x13 = 5.68692405463212
x14 = -94.8440408602413
x15 = -103.076296315916
x16 = 11.9701093618117
x17 = -13.1626318669066
x18 = 37.1028505905301
x19 = 24.5364799761709
x20 = 81.0851477407872
x21 = -96.7931110087361
x22 = 60.2865216707535
x23 = -77.9435550871974
x24 = -258.206858846911
x25 = 62.2355918192484
x26 = -0.596261252547466
x27 = 30.8196652833505
x28 = 85.4192628994719
x29 = -59.0939991656586
x30 = 28.8705951348556
x31 = 3.73785390613726
x32 = -90.5099257015565
x33 = -65.3771844728382
x34 = 22.587409827676
x35 = -207.941376389474
x36 = 72.8528922851127
x37 = 10.0210392133168
x38 = 91.7024482066515
x39 = -32.0121877884454
x40 = -25.7290024812658
x41 = 79.1360775922923
x42 = -33.9612579369403
x43 = -50.8617437099842
x44 = -40.2444432441198
x45 = 18.2532946689913
x46 = 35.1537804420352
x47 = -84.2267403943769
x48 = -2.54533140104233
x49 = -75.9944849387025
x50 = 97.9856335138311
x51 = 55.9524065120688
x52 = -6.87944655972705
x53 = -38.295373095625
x54 = -27.6780726297607
x55 = -52.810813858479
x56 = 41.4369657492148
x57 = 93.6515183551463
x58 = -15.1117020154015
x59 = 16.3042245204964
x60 = -82.2776702458821
x61 = -19.4458171740862
x62 = 99.9347036623259
x63 = -57.1449290171637
x64 = 87.3683330479667
x65 = 74.8019624336076
x66 = 66.5697069779331
x67 = -63.4281143243433
x68 = -46.5276285512994
x69 = -44.5785584028046
x70 = -8.82851670822191
x70 = -8.82851670822191