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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
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} - 2 = 0$$
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} - 2 = 0$$
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
$$w_{1} = 1$$
$$w_{2} = 2$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$\left(3 \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) - 3 = 0$$
cambiamos
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} - 2 = 0$$
$$- \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} - 2 = 0$$
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) = 1
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = 1$$
$$w_{2} = 2$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
Respuesta rápida
[src]
pi
x1 = --
2
$$x_{1} = \frac{\pi}{2}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x2 = 2*re|atan|- - -------|| + 2*I*im|atan|- - -------||
\ \2 2 // \ \2 2 //
$$x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x3 = 2*re|atan|- + -------|| + 2*I*im|atan|- + -------||
\ \2 2 // \ \2 2 //
$$x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
x3 = 2*re(atan(1/2 + sqrt(3)*i/2)) + 2*i*im(atan(1/2 + sqrt(3)*i/2))
Suma y producto de raíces
[src]
suma
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ pi | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || -- + 2*re|atan|- - -------|| + 2*I*im|atan|- - -------|| + 2*re|atan|- + -------|| + 2*I*im|atan|- + -------|| 2 \ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\left(\frac{\pi}{2} + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ pi | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || -- + 2*re|atan|- + -------|| + 2*re|atan|- - -------|| + 2*I*im|atan|- + -------|| + 2*I*im|atan|- - -------|| 2 \ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\frac{\pi}{2} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
producto
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ pi | | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 ||| --*|2*re|atan|- - -------|| + 2*I*im|atan|- - -------|||*|2*re|atan|- + -------|| + 2*I*im|atan|- + -------||| 2 \ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$\frac{\pi}{2} \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
| | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 |||
2*pi*|I*im|atan|- + -------|| + re|atan|- + -------|||*|I*im|atan|- - -------|| + re|atan|- - -------|||
\ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$2 \pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
2*pi*(i*im(atan(1/2 + i*sqrt(3)/2)) + re(atan(1/2 + i*sqrt(3)/2)))*(i*im(atan(1/2 - i*sqrt(3)/2)) + re(atan(1/2 - i*sqrt(3)/2)))
Respuesta numérica
[src]
x1 = -42.4115000881114
x2 = 83.2522058481918
x3 = 76.9690207793905
x4 = 95.8185747883961
x5 = 64.4026506037314
x6 = 7.85398285538609
x7 = 51.8362789090115
x8 = -67.544240879025
x9 = -23.561946075942
x10 = 20.4203510568788
x11 = 64.4026493044641
x12 = 51.8362799897705
x13 = -73.8274272794653
x14 = 14.1371656591617
x15 = 64.4026481915252
x16 = -36.1283142806347
x17 = -36.1283166952282
x18 = 45.553094091839
x19 = -17.2787598788452
x20 = 1.57079785005069
x21 = -80.1106125755117
x22 = 1.57079536523077
x23 = -23.561945016053
x24 = 7.85398046563447
x25 = 70.6858340517028
x26 = 76.9690195526133
x27 = 26.703537282924
x28 = -98.9601678826108
x29 = -67.5442421763137
x30 = -80.1106114181945
x31 = -42.4115005430641
x32 = 89.5353909435736
x33 = -10.9955747752993
x34 = -48.6946856448184
x35 = 39.2699086837397
x36 = -86.3937971842945
x37 = -92.6769845303487
x38 = -92.676982808917
x39 = -17.2787601164358
x40 = 32.9867223887206
x41 = -117.80972560988
x42 = -48.6946868672216
x43 = 70.6858344445529
x44 = 1.57079661901596
x45 = 26.7035369653861
x46 = 26.7035385469741
x47 = -10.9955735516589
x48 = 58.1194628121746
x49 = -61.2610571936019
x50 = 102.101759965899
x51 = 83.2522046289214
x52 = -92.6769840326577
x53 = 95.8185771224127
x54 = 14.1371671181822
x55 = 95.8185760701987
x56 = 83.2522058456645
x57 = -36.1283154137715
x58 = 45.5530925300164
x59 = 58.1194653976648
x60 = 1.57079700398873
x61 = -54.9778719394428
x62 = -73.8274260609448
x63 = -29.8451300938139
x64 = 14.1371682454946
x65 = -17.2787586177095
x66 = 7.85398174770883
x67 = -86.3937989639545
x68 = 45.5530937812277
x69 = -325.154840065363
x70 = 58.1194643770702
x71 = 32.9867236138576
x72 = -4.71238848059836
x73 = -98.9601691037059
x74 = 51.8362776268483
x75 = -61.2610570407565
x76 = -67.544243206816
x77 = -80.1106138557219
x78 = -73.8274286445858
x79 = 89.5353911752829
x80 = -61.2610557825211
x81 = 39.2699074635758
x82 = -23.5619437177603
x83 = 20.4203521441984
x84 = -4.71238970180774
x85 = -54.9778707171509
x86 = 70.6858357115182
x87 = -86.3937977050157
x88 = 20.4203534431639
x89 = 89.5353896949152
x90 = -42.4115017994301
x91 = -29.8451289073854
x92 = -29.8451314931042
x92 = -29.8451314931042
Gráfico