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cos(x)^(2)+3*sin(x)+3=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 cos (x) + 3*sin(x) + 3 = 0
$$\left(3 \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + 3 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(3 \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + 3 = 0$$
cambiamos
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} + 4 = 0$$
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} + 4 = 0$$
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
$$a = -1$$
$$b = 3$$
$$c = 4$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = -1$$
$$w_{2} = 4$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(4 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(4 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(4 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(4 \right)}$$
$$\left(3 \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + 3 = 0$$
cambiamos
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} + 4 = 0$$
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} + 4 = 0$$
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
$$a = -1$$
$$b = 3$$
$$c = 4$$
, entonces
D = b^2 - 4 * a * c =
(3)^2 - 4 * (-1) * (4) = 25
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = -1$$
$$w_{2} = 4$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(4 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(4 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(4 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(4 \right)}$$
Suma y producto de raíces
[src]
suma
/ / ____\\ / / ____\\ / / ____\\ / / ____\\ pi | |1 I*\/ 15 || | |1 I*\/ 15 || | |1 I*\/ 15 || | |1 I*\/ 15 || - -- + 2*re|atan|- - --------|| + 2*I*im|atan|- - --------|| + 2*re|atan|- + --------|| + 2*I*im|atan|- + --------|| 2 \ \4 4 // \ \4 4 // \ \4 4 // \ \4 4 //
$$\left(- \frac{\pi}{2} + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)}\right)$$
=
/ / ____\\ / / ____\\ / / ____\\ / / ____\\
| |1 I*\/ 15 || | |1 I*\/ 15 || pi | |1 I*\/ 15 || | |1 I*\/ 15 ||
2*re|atan|- - --------|| + 2*re|atan|- + --------|| - -- + 2*I*im|atan|- - --------|| + 2*I*im|atan|- + --------||
\ \4 4 // \ \4 4 // 2 \ \4 4 // \ \4 4 //
$$- \frac{\pi}{2} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)}$$
producto
/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ -pi | | |1 I*\/ 15 || | |1 I*\/ 15 ||| | | |1 I*\/ 15 || | |1 I*\/ 15 ||| ----*|2*re|atan|- - --------|| + 2*I*im|atan|- - --------|||*|2*re|atan|- + --------|| + 2*I*im|atan|- + --------||| 2 \ \ \4 4 // \ \4 4 /// \ \ \4 4 // \ \4 4 ///
$$- \frac{\pi}{2} \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)}\right)$$
=
/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\
| | |1 I*\/ 15 || | |1 I*\/ 15 ||| | | |1 I*\/ 15 || | |1 I*\/ 15 |||
-2*pi*|I*im|atan|- - --------|| + re|atan|- - --------|||*|I*im|atan|- + --------|| + re|atan|- + --------|||
\ \ \4 4 // \ \4 4 /// \ \ \4 4 // \ \4 4 ///
$$- 2 \pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)}\right)$$
-2*pi*(i*im(atan(1/4 - i*sqrt(15)/4)) + re(atan(1/4 - i*sqrt(15)/4)))*(i*im(atan(1/4 + i*sqrt(15)/4)) + re(atan(1/4 + i*sqrt(15)/4)))
Respuesta rápida
[src]
-pi
x1 = ----
2
$$x_{1} = - \frac{\pi}{2}$$
/ / ____\\ / / ____\\
| |1 I*\/ 15 || | |1 I*\/ 15 ||
x2 = 2*re|atan|- - --------|| + 2*I*im|atan|- - --------||
\ \4 4 // \ \4 4 //
$$x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} - \frac{\sqrt{15} i}{4} \right)}\right)}$$
/ / ____\\ / / ____\\
| |1 I*\/ 15 || | |1 I*\/ 15 ||
x3 = 2*re|atan|- + --------|| + 2*I*im|atan|- + --------||
\ \4 4 // \ \4 4 //
$$x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{15} i}{4} \right)}\right)}$$
x3 = 2*re(atan(1/4 + sqrt(15)*i/4)) + 2*i*im(atan(1/4 + sqrt(15)*i/4))
Respuesta numérica
[src]
x1 = 2122.14583792168
x2 = -58.1194639983404
x3 = -26.7035379113368
x4 = -14.137166267618
x5 = -20.4203520160774
x6 = 36.1283157093032
x7 = -158.650429510901
x8 = 4.71238930682327
x9 = -45.5530933379328
x10 = 80.110613139437
x11 = -58.119464108743
x12 = 61.2610563902501
x13 = -32.9867225012491
x14 = 86.3937980345907
x15 = -70.6858343897312
x16 = -95.8185764895439
x17 = 48.6946859098066
x18 = -32.9867231822668
x19 = -45.5530935900185
x20 = -39.2699084040555
x21 = -76.9690203364266
x22 = -70.6858350616861
x23 = -76.9690194223839
x24 = 86.3937978874973
x25 = -89.5353907488532
x26 = -76.9690196523847
x27 = -64.4026497009475
x28 = 54.9778711266118
x29 = -95.8185758680815
x30 = -39.2699078379722
x31 = -51.8362786894957
x32 = -1.57079621145266
x33 = -51.8362794146821
x34 = -7.85398149740641
x35 = -7.85398233977009
x36 = 17.2787592393122
x37 = -20.4203525581159
x38 = 67.5442422976148
x39 = 92.6769830669358
x40 = -83.2522056059716
x41 = 98.9601682813222
x42 = 73.8274274816964
x43 = 73.8274272534934
x44 = 10.9955739720808
x45 = -89.5353904664517
x46 = 61.2610570733106
x47 = -227.765467380722
x48 = 98.9601689505548
x49 = 54.9778718002295
x50 = -83.2522049833183
x51 = -26.7035372352027
x52 = 29.8451301301352
x53 = -14.1371669906886
x54 = 29.8451303220427
x55 = -114.668132439338
x56 = 17.2787599191456
x57 = -64.402649173144
x58 = -83.2522055607843
x59 = 23.5619451408423
x60 = 92.676983594607
x61 = -1.57079643102431
x62 = 48.6946864510303
x63 = 23.5619445854305
x64 = 42.4115009127929
x65 = 42.411500728142
x66 = 67.5442417296428
x67 = -14.1371668379482
x68 = -32.9867229016282
x69 = 4.71238875280883
x70 = -32.9867229219624
x71 = 10.9955746496251
x71 = 10.9955746496251