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Expresiones semejantes
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sin^2(x)-2cos^2(x)-1/2=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 1
sin (x) - 2*cos (x) - - = 0
2
$$\left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) - \frac{1}{2} = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) - \frac{1}{2} = 0$$
cambiamos
$$3 \sin^{2}{\left(x \right)} - \frac{5}{2} = 0$$
$$3 \sin^{2}{\left(x \right)} - \frac{5}{2} = 0$$
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
$$a = 3$$
$$b = 0$$
$$c = - \frac{5}{2}$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{\sqrt{30}}{6}$$
$$w_{2} = - \frac{\sqrt{30}}{6}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{30}}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{30}}{6} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)} + \pi$$
$$\left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) - \frac{1}{2} = 0$$
cambiamos
$$3 \sin^{2}{\left(x \right)} - \frac{5}{2} = 0$$
$$3 \sin^{2}{\left(x \right)} - \frac{5}{2} = 0$$
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
$$a = 3$$
$$b = 0$$
$$c = - \frac{5}{2}$$
, entonces
D = b^2 - 4 * a * c =
(0)^2 - 4 * (3) * (-5/2) = 30
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{\sqrt{30}}{6}$$
$$w_{2} = - \frac{\sqrt{30}}{6}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{30}}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{30}}{6} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{30}}{6} \right)} + \pi$$
Respuesta rápida
[src]
/ _____________\
| ___ / ___ |
|\/ 5 *\/ 7 - 2*\/ 6 |
x1 = -2*atan|----------------------|
\ 5 /
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{7 - 2 \sqrt{6}}}{5} \right)}$$
/ _____________\
| ___ / ___ |
|\/ 5 *\/ 7 - 2*\/ 6 |
x2 = 2*atan|----------------------|
\ 5 /
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{7 - 2 \sqrt{6}}}{5} \right)}$$
/ _____________\
| ___ / ___ |
|\/ 5 *\/ 7 + 2*\/ 6 |
x3 = -2*atan|----------------------|
\ 5 /
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{2 \sqrt{6} + 7}}{5} \right)}$$
/ _____________\
| ___ / ___ |
|\/ 5 *\/ 7 + 2*\/ 6 |
x4 = 2*atan|----------------------|
\ 5 /
$$x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{2 \sqrt{6} + 7}}{5} \right)}$$
x4 = 2*atan(sqrt(5)*sqrt(2*sqrt(6) + 7)/5)
Suma y producto de raíces
[src]
suma
/ _____________\ / _____________\ / _____________\ / _____________\
| ___ / ___ | | ___ / ___ | | ___ / ___ | | ___ / ___ |
|\/ 5 *\/ 7 - 2*\/ 6 | |\/ 5 *\/ 7 - 2*\/ 6 | |\/ 5 *\/ 7 + 2*\/ 6 | |\/ 5 *\/ 7 + 2*\/ 6 |
- 2*atan|----------------------| + 2*atan|----------------------| - 2*atan|----------------------| + 2*atan|----------------------|
\ 5 / \ 5 / \ 5 / \ 5 /
$$\left(- 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{2 \sqrt{6} + 7}}{5} \right)} + \left(- 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{7 - 2 \sqrt{6}}}{5} \right)} + 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{7 - 2 \sqrt{6}}}{5} \right)}\right)\right) + 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{2 \sqrt{6} + 7}}{5} \right)}$$
=
0
$$0$$
producto
/ _____________\ / _____________\ / _____________\ / _____________\
| ___ / ___ | | ___ / ___ | | ___ / ___ | | ___ / ___ |
|\/ 5 *\/ 7 - 2*\/ 6 | |\/ 5 *\/ 7 - 2*\/ 6 | |\/ 5 *\/ 7 + 2*\/ 6 | |\/ 5 *\/ 7 + 2*\/ 6 |
-2*atan|----------------------|*2*atan|----------------------|*-2*atan|----------------------|*2*atan|----------------------|
\ 5 / \ 5 / \ 5 / \ 5 /
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{7 - 2 \sqrt{6}}}{5} \right)} 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{7 - 2 \sqrt{6}}}{5} \right)} \left(- 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{2 \sqrt{6} + 7}}{5} \right)}\right) 2 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{2 \sqrt{6} + 7}}{5} \right)}$$
=
/ _____________\ / _____________\
| ___ / ___ | | ___ / ___ |
2|\/ 5 *\/ 7 - 2*\/ 6 | 2|\/ 5 *\/ 7 + 2*\/ 6 |
16*atan |----------------------|*atan |----------------------|
\ 5 / \ 5 /
$$16 \operatorname{atan}^{2}{\left(\frac{\sqrt{5} \sqrt{7 - 2 \sqrt{6}}}{5} \right)} \operatorname{atan}^{2}{\left(\frac{\sqrt{5} \sqrt{2 \sqrt{6} + 7}}{5} \right)}$$
16*atan(sqrt(5)*sqrt(7 - 2*sqrt(6))/5)^2*atan(sqrt(5)*sqrt(7 + 2*sqrt(6))/5)^2
Respuesta numérica
[src]
x1 = -70.2653003704864
x2 = 32.5661885274089
x3 = -41.9909664881782
x4 = 92.2564489456149
x5 = -76.548485677666
x6 = 52.2568131195156
x7 = -61.6815910802849
x8 = 38.8493738345884
x9 = 54.5573371025374
x10 = -33.4072571979768
x11 = -32.5661885274089
x12 = -48.2741517953578
x13 = -96.2391102697727
x14 = -89.9559249625931
x15 = -83.6727396554135
x16 = -98.5396342527945
x17 = 89.9559249625931
x18 = 11.4161086228482
x19 = 5.13292331566865
x20 = 55.3984057731053
x21 = -79.6900783312558
x22 = 96.2391102697727
x23 = 45.973627812336
x24 = 60.840522409717
x25 = 16.8582252594599
x26 = 291.01785479234
x27 = 1.99133066207886
x28 = -27.1240718907972
x29 = -52.2568131195156
x30 = -17.6992939300278
x31 = 82.8316709848456
x32 = 23.9824792372074
x33 = 26.2830032202293
x34 = -64.8231837338747
x35 = 28064.9974365091
x36 = 10.5750399522803
x37 = -211.636969782027
x38 = 33.4072571979768
x39 = 63.9821150633068
x40 = -49.1152204659258
x41 = 8.27451596925845
x42 = -23.9824792372074
x43 = -67.9647763874645
x44 = -74.2479616946441
x45 = 99.3807029233624
x46 = -73.4068930240762
x47 = -10.5750399522803
x48 = -99.3807029233624
x49 = 85.9732636384354
x50 = 70.2653003704864
x51 = 253.318742949262
x52 = 4.29185464510072
x53 = 83.6727396554135
x54 = -4.29185464510072
x55 = 7648.62784949964
x56 = 19.9998179130497
x57 = -45.973627812336
x58 = -1.99133066207886
x59 = -13.7166326058701
x60 = 48.2741517953578
x61 = -93.0975176161829
x62 = 76.548485677666
x63 = -71.1063690410543
x64 = 98.5396342527945
x65 = -19.9998179130497
x66 = 61.6815910802849
x67 = -8.27451596925845
x68 = -26.2830032202293
x69 = -57.6989297561272
x70 = -54.5573371025374
x71 = -55.3984057731053
x72 = -77.3895543482339
x73 = 30.265664544387
x74 = -63.9821150633068
x75 = 133.097153442282
x76 = -5.13292331566865
x77 = 17.6992939300278
x78 = -35.7077811809987
x79 = 74.2479616946441
x80 = 67.9647763874645
x81 = -85.9732636384354
x82 = 39.6904425051564
x83 = -39.6904425051564
x84 = -11.4161086228482
x85 = -92.2564489456149
x86 = 41.9909664881782
x87 = 77.3895543482339
x87 = 77.3895543482339