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2cos^2(x)+(sin(x)-1)(2-sqrt(3))=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 / ___\ 2*cos (x) + (sin(x) - 1)*\2 - \/ 3 / = 0
$$\left(2 - \sqrt{3}\right) \left(\sin{\left(x \right)} - 1\right) + 2 \cos^{2}{\left(x \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$\left(2 - \sqrt{3}\right) \left(\sin{\left(x \right)} - 1\right) + 2 \cos^{2}{\left(x \right)} = 0$$
cambiamos
$$\left(2 - \sqrt{3}\right) \left(\sin{\left(x \right)} - 1\right) - 2 \sin^{2}{\left(x \right)} + 2 = 0$$
$$\left(2 - \sqrt{3}\right) \left(\sin{\left(x \right)} - 1\right) - 2 \sin^{2}{\left(x \right)} + 2 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Abramos la expresión en la ecuación
$$- 2 w^{2} + \left(2 - \sqrt{3}\right) \left(w - 1\right) + 2 = 0$$
Obtenemos la ecuación cuadrática
$$- 2 w^{2} - \sqrt{3} w + 2 w + \sqrt{3} = 0$$
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
$$a = -2$$
$$b = 2 - \sqrt{3}$$
$$c = \sqrt{3}$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{2}$$
$$w_{2} = - \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} + \frac{\sqrt{3}}{4} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} + \frac{\sqrt{3}}{4} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)} + \pi$$
$$\left(2 - \sqrt{3}\right) \left(\sin{\left(x \right)} - 1\right) + 2 \cos^{2}{\left(x \right)} = 0$$
cambiamos
$$\left(2 - \sqrt{3}\right) \left(\sin{\left(x \right)} - 1\right) - 2 \sin^{2}{\left(x \right)} + 2 = 0$$
$$\left(2 - \sqrt{3}\right) \left(\sin{\left(x \right)} - 1\right) - 2 \sin^{2}{\left(x \right)} + 2 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Abramos la expresión en la ecuación
$$- 2 w^{2} + \left(2 - \sqrt{3}\right) \left(w - 1\right) + 2 = 0$$
Obtenemos la ecuación cuadrática
$$- 2 w^{2} - \sqrt{3} w + 2 w + \sqrt{3} = 0$$
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
$$a = -2$$
$$b = 2 - \sqrt{3}$$
$$c = \sqrt{3}$$
, entonces
D = b^2 - 4 * a * c =
(2 - sqrt(3))^2 - 4 * (-2) * (sqrt(3)) = (2 - sqrt(3))^2 + 8*sqrt(3)
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{2}$$
$$w_{2} = - \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} + \frac{\sqrt{3}}{4} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} + \frac{\sqrt{3}}{4} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{4} + \frac{1}{2} + \frac{\sqrt{\left(2 - \sqrt{3}\right)^{2} + 8 \sqrt{3}}}{4} \right)} + \pi$$
Suma y producto de raíces
[src]
suma
2*pi pi pi - ---- - -- + -- 3 3 2
$$\left(- \frac{2 \pi}{3} - \frac{\pi}{3}\right) + \frac{\pi}{2}$$
=
-pi ---- 2
$$- \frac{\pi}{2}$$
producto
-2*pi -pi pi -----*----*-- 3 3 2
$$\frac{\pi}{2} \cdot - \frac{2 \pi}{3} \left(- \frac{\pi}{3}\right)$$
=
3 pi --- 9
$$\frac{\pi^{3}}{9}$$
pi^3/9
Respuesta rápida
[src]
-2*pi
x1 = -----
3
$$x_{1} = - \frac{2 \pi}{3}$$
-pi
x2 = ----
3
$$x_{2} = - \frac{\pi}{3}$$
pi
x3 = --
2
$$x_{3} = \frac{\pi}{2}$$
x3 = pi/2
Respuesta numérica
[src]
x1 = -36.128315419384
x2 = -48.694685882713
x3 = 45.5530936981186
x4 = -39.7935069454707
x5 = -42.4115006118114
x6 = -10.9955742707118
x7 = 4.18879020478639
x8 = 76.9690199200916
x9 = 54.4542726622231
x10 = -86.3937977671983
x11 = 24.0855436775217
x12 = -98.9601688162366
x13 = -23.5619450088582
x14 = -92.6769830320043
x15 = 17.8023583703422
x16 = 11.5191730631626
x17 = 1.57079654329663
x18 = -92.6769829855642
x19 = -57.5958653158129
x20 = 70.6858345033647
x21 = 41.8879020478639
x22 = 61.7846555205993
x23 = -4.71238873402792
x24 = -33.5103216382911
x25 = 49.2182849062401
x26 = -54.9778716704722
x27 = 7.85398174040168
x28 = 16.7551608191456
x29 = 51.8362791242123
x30 = 26.7035373480551
x31 = -98.9601684973165
x32 = 7.85398168599725
x33 = -26.1799387799149
x34 = 32.9867228301953
x35 = 32.9867226238939
x36 = -132.994089001968
x37 = -32.4631240870945
x38 = 89.5353908526462
x39 = 14.1371671044393
x40 = 85.870199198121
x41 = 60.7374579694027
x42 = -77.4926187885482
x43 = -46.0766922526503
x44 = -54.9778713804668
x45 = 10.471975511966
x46 = 20.4203523182499
x47 = -83.7758040957278
x48 = 51.8362788997593
x49 = -13.6135681655558
x50 = -80.1106125796694
x51 = 83.2522055612909
x52 = -19.8967534727354
x53 = 55.5014702134197
x54 = -58.6430628670095
x55 = 64.4026493088016
x56 = 76.9690197722979
x57 = -63.8790506229925
x58 = -27.2271363311115
x59 = 95.8185760587403
x60 = 30.3687289847013
x61 = 92.1533845053006
x62 = 74.3510261349584
x63 = 48.1710873550435
x64 = -4.71238886155635
x65 = -2.0943951023932
x66 = 99.4837673636768
x67 = -90.0589894029074
x68 = 39.2699082235616
x69 = -10.9955745237262
x70 = 20.4203521498545
x71 = -54.977871366858
x72 = 83.2522053206189
x73 = -48.694685932077
x74 = -89.0117918517108
x75 = 98.4365698124802
x76 = -29.8451300964315
x77 = -76.4454212373516
x78 = -61.2610569617241
x79 = -73.8274272800547
x80 = -17.2787598069783
x81 = 39.2699084139954
x82 = 58.1194644436463
x83 = 68.0678408277789
x84 = -96.342174710087
x85 = -71.2094334813686
x86 = -67.5442421673472
x87 = -70.162235930172
x87 = -70.162235930172