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2sin^2(1/2 x)+C=-2cos(x/2)sin(x/2)+C

Solución

2 sin^{2} (\frac{1}{2} x) + C = - 2 cos (\frac{x}{2}) sin (\frac{x}{2}) + C

Solución

x = 4 πn, x = 2 π + 4 πn, x = \frac{3 π}{2} + 2 πn

Grados

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Pasos de solución

2 sin^{2} (\frac{1}{2} x) + C = - 2 cos (\frac{x}{2}) sin (\frac{x}{2}) + C

Restar

- 2 cos (\frac{x}{2}) sin (\frac{x}{2}) + C

de ambos lados

2 sin^{2} (\frac{x}{2}) + 2 cos (\frac{x}{2}) sin (\frac{x}{2}) = 0

Factorizar

2 sin^{2} (\frac{x}{2}) + 2 cos (\frac{x}{2}) sin (\frac{x}{2}) : 2 sin (\frac{x}{2}) (sin (\frac{x}{2}) + cos (\frac{x}{2}))

2 sin^{2} (\frac{x}{2}) + 2 cos (\frac{x}{2}) sin (\frac{x}{2})

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

sin^{2} (\frac{x}{2}) = sin (\frac{x}{2}) sin (\frac{x}{2})

= 2 sin (\frac{x}{2}) sin (\frac{x}{2}) + 2 sin (\frac{x}{2}) cos (\frac{x}{2})

Factorizar el termino común

2 sin (\frac{x}{2})

= 2 sin (\frac{x}{2}) (sin (\frac{x}{2}) + cos (\frac{x}{2}))

2 sin (\frac{x}{2}) (sin (\frac{x}{2}) + cos (\frac{x}{2})) = 0

Resolver cada parte por separado

sin (\frac{x}{2}) = 0 or sin (\frac{x}{2}) + cos (\frac{x}{2}) = 0

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

sin (\frac{x}{2}) = 0

Soluciones generales para

sin (\frac{x}{2}) = 0

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

Resolver

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

Multiplicar ambos lados por

2

\frac{x}{2} = 2 πn

Multiplicar ambos lados por

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplificar

x = 4 πn

x = 4 πn

Resolver

\frac{x}{2} = π + 2 πn : x = 2 π + 4 πn

\frac{x}{2} = π + 2 πn

Multiplicar ambos lados por

2

\frac{x}{2} = π + 2 πn

Multiplicar ambos lados por

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplificar

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

sin (\frac{x}{2}) + cos (\frac{x}{2}) = 0 : x = \frac{3 π}{2} + 2 πn

sin (\frac{x}{2}) + cos (\frac{x}{2}) = 0

Re-escribir usando identidades trigonométricas

sin (\frac{x}{2}) + cos (\frac{x}{2}) = 0

Dividir ambos lados entre

cos (\frac{x}{2}), cos (\frac{x}{2}) \neq = 0

\frac{sin ( \frac{x}{2} ) + cos ( \frac{x}{2} )}{cos ( \frac{x}{2} )} = \frac{0}{cos ( \frac{x}{2} )}

Simplificar

\frac{sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )} + 1 = 0

Utilizar la identidad trigonométrica básica:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (\frac{x}{2}) + 1 = 0

tan (\frac{x}{2}) + 1 = 0

Mover

1

al lado derecho

tan (\frac{x}{2}) + 1 = 0

Restar

1

de ambos lados

tan (\frac{x}{2}) + 1 - 1 = 0 - 1

Simplificar

tan (\frac{x}{2}) = - 1

tan (\frac{x}{2}) = - 1

Soluciones generales para

tan (\frac{x}{2}) = - 1

tan (x)

tabla de valores periódicos con

πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

\frac{x}{2} = \frac{3 π}{4} + πn

\frac{x}{2} = \frac{3 π}{4} + πn

Resolver

\frac{x}{2} = \frac{3 π}{4} + πn : x = \frac{3 π}{2} + 2 πn

\frac{x}{2} = \frac{3 π}{4} + πn

Multiplicar ambos lados por

2

\frac{x}{2} = \frac{3 π}{4} + πn

Multiplicar ambos lados por

2

\frac{2 x}{2} = 2 \cdot \frac{3 π}{4} + 2 πn

Simplificar

\frac{2 x}{2} = 2 \cdot \frac{3 π}{4} + 2 πn

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

2 \cdot \frac{3 π}{4} + 2 πn : \frac{3 π}{2} + 2 πn

2 \cdot \frac{3 π}{4} + 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 π 2}{4}

Multiplicar los numeros:

3 \cdot 2 = 6

= \frac{6 π}{4}

Eliminar los terminos comunes:

2

= \frac{3 π}{2}

= \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combinar toda las soluciones

x = 4 πn, x = 2 π + 4 πn, x = \frac{3 π}{2} + 2 πn

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