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Expresiones idénticas
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Expresiones semejantes
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- Seno sin
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- sin(2x)+sin(x)=0
- sin^2(2x)-sqrt(3)*sin(2*x)+2-cos^2(x)-sin(x)=0
3*cos^2(x)-sin(x)-1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 3*cos (x) - sin(x) - 1 = 0
$$\left(- \sin{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) - 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(- \sin{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) - 1 = 0$$
cambiamos
$$- 3 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 2 = 0$$
$$- 3 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 2 = 0$$
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
$$a = -3$$
$$b = -1$$
$$c = 2$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = -1$$
$$w_{2} = \frac{2}{3}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{3} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{3} \right)} + \pi$$
$$\left(- \sin{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) - 1 = 0$$
cambiamos
$$- 3 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 2 = 0$$
$$- 3 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 2 = 0$$
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
$$a = -3$$
$$b = -1$$
$$c = 2$$
, entonces
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (-3) * (2) = 25
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = -1$$
$$w_{2} = \frac{2}{3}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{3} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{3} \right)} + \pi$$
Respuesta rápida
[src]
-pi
x1 = ----
2
$$x_{1} = - \frac{\pi}{2}$$
/ ___\
|3 \/ 5 |
x2 = 2*atan|- - -----|
\2 2 /
$$x_{2} = 2 \operatorname{atan}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}$$
/ ___\
|3 \/ 5 |
x3 = 2*atan|- + -----|
\2 2 /
$$x_{3} = 2 \operatorname{atan}{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}$$
x3 = 2*atan(sqrt(5)/2 + 3/2)
Suma y producto de raíces
[src]
suma
/ ___\ / ___\ pi |3 \/ 5 | |3 \/ 5 | - -- + 2*atan|- - -----| + 2*atan|- + -----| 2 \2 2 / \2 2 /
$$\left(- \frac{\pi}{2} + 2 \operatorname{atan}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}\right) + 2 \operatorname{atan}{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}$$
=
/ ___\ / ___\
|3 \/ 5 | |3 \/ 5 | pi
2*atan|- + -----| + 2*atan|- - -----| - --
\2 2 / \2 2 / 2
$$- \frac{\pi}{2} + 2 \operatorname{atan}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)} + 2 \operatorname{atan}{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}$$
producto
/ ___\ / ___\ -pi |3 \/ 5 | |3 \/ 5 | ----*2*atan|- - -----|*2*atan|- + -----| 2 \2 2 / \2 2 /
$$- \frac{\pi}{2} \cdot 2 \operatorname{atan}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)} 2 \operatorname{atan}{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}$$
=
/ ___\ / ___\
|3 \/ 5 | |3 \/ 5 |
-2*pi*atan|- + -----|*atan|- - -----|
\2 2 / \2 2 /
$$- 2 \pi \operatorname{atan}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)} \operatorname{atan}{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}$$
-2*pi*atan(3/2 + sqrt(5)/2)*atan(3/2 - sqrt(5)/2)
Respuesta numérica
[src]
x1 = -68.3853107227485
x2 = 0.729727656226966
x3 = 63.5615807280228
x4 = 80.1106131448878
x5 = 25.8624688849453
x6 = 42.4115007295237
x7 = -51.8362786898351
x8 = 71.5269033763383
x9 = -58.1194639996762
x10 = 46.3941621476199
x11 = -5.55345765095262
x12 = -47.8536174600739
x13 = -62.1021254155689
x14 = -76.9690201113014
x15 = -64.4026491927816
x16 = -93.5180519514668
x17 = -14.1371668397229
x18 = 73.8274274788253
x19 = 88.6943219567412
x20 = 27.5446062260812
x21 = 19.5792835777657
x22 = -87.2348666442873
x23 = 98.9601683638967
x24 = 54.9778712172624
x25 = -70.6858344735403
x26 = -32.9867230725928
x27 = 67.5442422708464
x28 = -99.8012372586464
x29 = -55.8189401083893
x30 = 44.7120248064841
x31 = -70.685834226677
x32 = 61.2610569654502
x33 = -1.570796429238
x34 = 52.6773474547995
x35 = 96.6596446050566
x36 = 29.8451303197576
x37 = -154.667767682127
x38 = -98.1190999175106
x39 = -18.1198282653118
x40 = 4.71238877421108
x41 = -70.685834863799
x42 = -43.2525694940301
x43 = 84.0932739906975
x44 = -1.57079590671519
x45 = -79.2695439959718
x46 = -91.835914610331
x47 = -20.4203520379199
x48 = 38.4288394993045
x49 = 98.9601680673456
x50 = -85.5527293031514
x51 = 2.41186499736283
x52 = -26.7035373262939
x53 = -89.5353907460495
x54 = 32.1456541921249
x55 = 23.5619451166262
x56 = -26.7035369941492
x57 = -76.9690202154601
x58 = -41.5704321528943
x59 = -39.269908380233
x60 = 36.1283160257602
x61 = 17.2787598200741
x62 = 77.8100886835179
x63 = 69.8447660352024
x64 = 48.694685928977
x65 = 17.2787600390192
x66 = -16.4376909241759
x67 = 92.6769830840502
x68 = -49.5357548012097
x69 = -32.9867230407201
x70 = 10.9955740717132
x71 = 54.9778710429089
x72 = 249.756616376998
x73 = 61.261057064485
x74 = -83.252205014109
x75 = -60.419988074433
x76 = -35.2872468457147
x77 = -3.87132030981676
x78 = 82.4111366495616
x79 = -11.8366429581322
x80 = -24.4030135724914
x81 = -83.252205534339
x82 = 10.9955740108644
x83 = 86.3937978885372
x84 = -45.5530935877598
x85 = -95.8185758681445
x86 = 8.69505030454241
x87 = 90.376459297877
x88 = -10.1545056169963
x89 = 40.1109768404403
x90 = 76.127951342382
x91 = -7.85398149897218
x92 = 33.8277915332608
x93 = -54.1368027672534
x94 = -39.2699081535158
x94 = -39.2699081535158