sin(x/2)+cos(x)-1=0

Solución

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

x = 4 πn, x = 2 π + 4 πn, x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

Pasos de solución

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

Sea:

u = \frac{x}{2}

sin (u) + cos (2 u) - 1 = 0

Re-escribir usando identidades trigonométricas

sin (u) - 2 sin^{2} (u) = 0

Usando el método de sustitución

sin (u) = 0, sin (u) = \frac{1}{2}

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = \frac{1}{2} : u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn

Combinar toda las soluciones

u = 2 πn, u = π + 2 πn, u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn

Sustituir en la ecuación

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 πn

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

\frac{x}{2} = \frac{π}{6} + 2 πn : x = \frac{π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{3} + 4 πn

x = 4 πn, x = 2 π + 4 πn, x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

3 sin (x) sec (x) = 2 sin (x)

y, x = sin (y)

2 sin (\frac{x}{2}) + 3 = 0

2 cos^{2} (x) + 3 sin (x) = 0

tan (3 x) - 1 = 0