2sin(2x)=3tan(x)

Solución

2 sin (2 x) = 3 tan (x)

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Pasos de solución

2 sin (2 x) = 3 tan (x)

Restar

3 tan (x)

de ambos lados

2 sin (2 x) - 3 tan (x) = 0

Expresar con seno, coseno

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 3 sin (x) + 2 cos (x) sin (2 x) = 0

Re-escribir usando identidades trigonométricas

- 3 sin (x) + 4 cos^{2} (x) sin (x) = 0

Factorizar

- 3 sin (x) + 4 cos^{2} (x) sin (x) : sin (x) (2 cos (x) + 3) (2 cos (x) - 3)

sin (x) (2 cos (x) + 3) (2 cos (x) - 3) = 0

Resolver cada parte por separado

sin (x) = 0 or 2 cos (x) + 3 = 0 or 2 cos (x) - 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

2 cos (x) + 3 = 0 : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

2 cos (x) - 3 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

tan (φ) = \frac{1}{3}

sin (θ) \cdot csc (1 8^{\circ}) = 1

tan^{2} (x) + csc^{2} (x) = 3

x, y = arctan (x + y)

cos (x) = - sin^{2} (x) - 1, 0 \leq x \leq 2 π