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solvefor a,sin(a+b/2)=cos(c/2)

Solución

resolver para

a, sin (a + \frac{b}{2}) = cos (\frac{c}{2})

Solución

a = \frac{π - c - b}{2} + 2 πn, a = π + 2 πn + \frac{- π + c - b}{2}

Pasos de solución

sin (a + \frac{b}{2}) = cos (\frac{c}{2})

Re-escribir usando identidades trigonométricas

cos (\frac{c}{2})

Usar la siguiente identidad:

cos (x) = sin (\frac{π}{2} - x)

sin (\frac{π}{2} - \frac{c}{2})

sin (a + \frac{b}{2}) = sin (\frac{π}{2} - \frac{c}{2})

Aplicar propiedades trigonométricas inversas

sin (a + \frac{b}{2}) = sin (\frac{π}{2} - \frac{c}{2})

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn, a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn, a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn : a = \frac{π - c - b}{2} + 2 πn

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn

Mover

\frac{b}{2}

al lado derecho

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn

Restar

\frac{b}{2}

de ambos lados

a + \frac{b}{2} - \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn - \frac{b}{2}

Simplificar

a + \frac{b}{2} - \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn - \frac{b}{2}

Simplificar

a + \frac{b}{2} - \frac{b}{2} : a

a + \frac{b}{2} - \frac{b}{2}

Sumar elementos similares:

\frac{b}{2} - \frac{b}{2} = 0

= a

Simplificar

\frac{π}{2} - \frac{c}{2} + 2 πn - \frac{b}{2} : \frac{π - c - b}{2} + 2 πn

\frac{π}{2} - \frac{c}{2} + 2 πn - \frac{b}{2}

Combinar las fracciones usando el mínimo común denominador:

\frac{π - c - b}{2}

Aplicar la regla

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

= \frac{π - c - b}{2}

= \frac{π - b - c}{2} + 2 πn

= \frac{π - c - b}{2} + 2 πn

a = \frac{π - c - b}{2} + 2 πn

a = \frac{π - c - b}{2} + 2 πn

a = \frac{π - c - b}{2} + 2 πn

a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn : a = π + 2 πn + \frac{- π + c - b}{2}

a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn

Mover

\frac{b}{2}

al lado derecho

a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn

Restar

\frac{b}{2}

de ambos lados

a + \frac{b}{2} - \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn - \frac{b}{2}

Simplificar

a + \frac{b}{2} - \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn - \frac{b}{2}

Simplificar

a + \frac{b}{2} - \frac{b}{2} : a

a + \frac{b}{2} - \frac{b}{2}

Sumar elementos similares:

\frac{b}{2} - \frac{b}{2} = 0

= a

Simplificar

π - (\frac{π}{2} - \frac{c}{2}) + 2 πn - \frac{b}{2} : π + 2 πn + \frac{- π + c - b}{2}

π - (\frac{π}{2} - \frac{c}{2}) + 2 πn - \frac{b}{2}

Combinar las fracciones usando el mínimo común denominador:

\frac{π - c}{2}

Aplicar la regla

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

= \frac{π - c}{2}

= π - \frac{- c + π}{2} + 2 πn - \frac{b}{2}

Quitar los parentesis:

(a) = a

= π - \frac{π - c}{2} + 2 πn - \frac{b}{2}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- ( π - c ) - b}{2}

- (π - c) : - π + c

- (π - c)

Poner los parentesis

= - π - (- c)

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

= - π + c

= π + 2 πn + \frac{c - b - π}{2}

a = π + 2 πn + \frac{- π + c - b}{2}

a = π + 2 πn + \frac{- π + c - b}{2}

a = π + 2 πn + \frac{- π + c - b}{2}

a = \frac{π - c - b}{2} + 2 πn, a = π + 2 πn + \frac{- π + c - b}{2}

Ejemplos populares

tan(x)-cot(x)=2cot^2(x)

tan (x) - cot (x) = 2 cot^{2} (x)

cot((-x+3n)/4)=0

cot (\frac{- x + 3 n}{4}) = 0

sin(x)+sin^2(x)+sin^3(x)=1

sin (x) + sin^{2} (x) + sin^{3} (x) = 1

(cos(a))/(1+sin(a))=sec(a)

\frac{cos ( a )}{1 + sin ( a )} = sec (a)

cos^2(x)+sin^2(2x)=0

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