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- La ecuación:
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Ecuación x+4=0
-
Ecuación 5x-4=7
-
Ecuación x^2+3x+2=0
- Ecuación x^2+y^2=0
- Expresar {x} en función de y en la ecuación:
- 18*x-11*y=2
- 5*x+4*y=-12
- 12*x-14*y=20
- -3*x-20*y=-13
- Derivada de:
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1/4
- Suma de la serie:
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1/4
- Integral de d{x}:
- 1/4
-
Expresiones idénticas
- sin(x)+cos(x)^(dos)= uno / cuatro
- seno de (x) más coseno de (x) en el grado (2) es igual a 1 dividir por 4
- seno de (x) más coseno de (x) en el grado (dos) es igual a uno dividir por cuatro
- sin(x)+cos(x)(2)=1/4
- sinx+cosx2=1/4
- sinx+cosx^2=1/4
- sin(x)+cos(x)^(2)=1 dividir por 4
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Expresiones semejantes
- sin(x)-cos(x)^(2)=1/4
- cos(pi*(x+1)/4)=2^(1/2)/2
- sinx+cosx^(2)=1/4
-
Expresiones con funciones
- Seno sin
- sin(x)=1,5
- sin(x-1)=0
- sin(x)=-√2/2
- sin(4*x)=1
- sin(-x)=1
- Coseno cos
- cos(x)=(-1/2)
- cos((pi*(x-7))/3)=1/2
- cos(x)/5=-1
- cos(2*x)+3*sin(x)-2=0
- cos(a-b)+sin(-a)sinb
sin(x)+cos(x)^(2)=1/4 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 sin(x) + cos (x) = 1/4
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} = \frac{1}{4}$$
Solución detallada
Tenemos la ecuación
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} = \frac{1}{4}$$
cambiamos
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{1}{4} = 0$$
$$- \sin^{2}{\left(x \right)} + \sin{\left(x \right)} + \frac{3}{4} = 0$$
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
$$a = -1$$
$$b = 1$$
$$c = \frac{3}{4}$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = \frac{3}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} \right)}$$
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} = \frac{1}{4}$$
cambiamos
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{1}{4} = 0$$
$$- \sin^{2}{\left(x \right)} + \sin{\left(x \right)} + \frac{3}{4} = 0$$
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
$$a = -1$$
$$b = 1$$
$$c = \frac{3}{4}$$
, entonces
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-1) * (3/4) = 4
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = \frac{3}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} \right)}$$
Suma y producto de raíces
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suma
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ 5*pi pi | |2 I*\/ 5 || | |2 I*\/ 5 || | |2 I*\/ 5 || | |2 I*\/ 5 || - ---- - -- + 2*re|atan|- - -------|| + 2*I*im|atan|- - -------|| + 2*re|atan|- + -------|| + 2*I*im|atan|- + -------|| 6 6 \ \3 3 // \ \3 3 // \ \3 3 // \ \3 3 //
$$\left(\left(- \frac{5 \pi}{6} - \frac{\pi}{6}\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |2 I*\/ 5 || | |2 I*\/ 5 || | |2 I*\/ 5 || | |2 I*\/ 5 ||
-pi + 2*re|atan|- - -------|| + 2*re|atan|- + -------|| + 2*I*im|atan|- - -------|| + 2*I*im|atan|- + -------||
\ \3 3 // \ \3 3 // \ \3 3 // \ \3 3 //
$$- \pi + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}$$
producto
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ -5*pi -pi | | |2 I*\/ 5 || | |2 I*\/ 5 ||| | | |2 I*\/ 5 || | |2 I*\/ 5 ||| -----*----*|2*re|atan|- - -------|| + 2*I*im|atan|- - -------|||*|2*re|atan|- + -------|| + 2*I*im|atan|- + -------||| 6 6 \ \ \3 3 // \ \3 3 /// \ \ \3 3 // \ \3 3 ///
$$- \frac{5 \pi}{6} \left(- \frac{\pi}{6}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
2 | | |2 I*\/ 5 || | |2 I*\/ 5 ||| | | |2 I*\/ 5 || | |2 I*\/ 5 |||
5*pi *|I*im|atan|- - -------|| + re|atan|- - -------|||*|I*im|atan|- + -------|| + re|atan|- + -------|||
\ \ \3 3 // \ \3 3 /// \ \ \3 3 // \ \3 3 ///
---------------------------------------------------------------------------------------------------------
9
$$\frac{5 \pi^{2} \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}\right)}{9}$$
5*pi^2*(i*im(atan(2/3 - i*sqrt(5)/3)) + re(atan(2/3 - i*sqrt(5)/3)))*(i*im(atan(2/3 + i*sqrt(5)/3)) + re(atan(2/3 + i*sqrt(5)/3)))/9
Respuesta rápida
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-5*pi
x1 = -----
6
$$x_{1} = - \frac{5 \pi}{6}$$
-pi
x2 = ----
6
$$x_{2} = - \frac{\pi}{6}$$
/ / ___\\ / / ___\\
| |2 I*\/ 5 || | |2 I*\/ 5 ||
x3 = 2*re|atan|- - -------|| + 2*I*im|atan|- - -------||
\ \3 3 // \ \3 3 //
$$x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)}$$
/ / ___\\ / / ___\\
| |2 I*\/ 5 || | |2 I*\/ 5 ||
x4 = 2*re|atan|- + -------|| + 2*I*im|atan|- + -------||
\ \3 3 // \ \3 3 //
$$x_{4} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}$$
x4 = 2*re(atan(2/3 + sqrt(5)*i/3)) + 2*i*im(atan(2/3 + sqrt(5)*i/3))
Respuesta numérica
[src]
x1 = -15.1843644923507
x2 = 18.3259571459405
x3 = 66.497044500984
x4 = 93.7241808320955
x5 = -57.0722665402146
x6 = -25.6563400043166
x7 = 56.025068989018
x8 = 47.6474885794452
x9 = -78.0162175641465
x10 = 5.75958653158129
x11 = -82.2050077689329
x12 = 28.7979326579064
x13 = 81.1578102177363
x14 = 97.9129710368819
x15 = -0.523598775598299
x16 = 68.5914396033772
x17 = -90.5825881785057
x18 = 43.4586983746588
x19 = 100.007366139275
x20 = -46.6002910282486
x21 = 30.8923277602996
x22 = 37.1755130674792
x23 = -63.3554518473942
x24 = -27.7507351067098
x25 = -34.0339204138894
x26 = -44.5058959258554
x27 = 79.0634151153431
x28 = 91.6297857297023
x29 = -65.4498469497874
x30 = 72.7802298081635
x31 = 60.2138591938044
x32 = -71.733032256967
x33 = -31.9395253114962
x34 = 53.9306738866248
x35 = 3.66519142918809
x36 = 22.5147473507269
x37 = -84.2994028713261
x38 = -88.4881930761125
x39 = -40.317105721069
x40 = -69.6386371545737
x41 = -50.789081233035
x42 = -6.80678408277789
x43 = -19.3731546971371
x44 = -103.148958792865
x45 = 87.4409955249159
x46 = 16.2315620435473
x47 = -38.2227106186758
x48 = 62.3082542961976
x49 = -94.7713783832921
x50 = 49.7418836818384
x51 = 85.3466004225227
x52 = 74.8746249105567
x53 = 12.0427718387609
x54 = -21.4675497995303
x55 = -59.1666616426078
x56 = -13.0899693899575
x57 = 41.3643032722656
x58 = 24.60914245312
x59 = 35.081117965086
x60 = -75.9218224617533
x61 = 9.94837673636768
x62 = -2.61799387799149
x62 = -2.61799387799149
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