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2sin(2x)=sin(x)

Solución

2 sin (2 x) = sin (x)

x = 2 πn, x = π + 2 πn, x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 75.52248 \dots^{\circ} + 36 0^{\circ} n, x = 284.47751 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

2 sin (2 x) = sin (x)

Restar

sin (x)

de ambos lados

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

Re-escribir usando identidades trigonométricas

- sin (x) + 2 sin (2 x)

Utilizar la identidad trigonométrica del ángulo doble:

sin (2 x) = 2 sin (x) cos (x)

= - sin (x) + 2 \cdot 2 sin (x) cos (x)

Simplificar

= - sin (x) + 4 sin (x) cos (x)

- sin (x) + 4 cos (x) sin (x) = 0

Factorizar

- sin (x) + 4 cos (x) sin (x) : sin (x) (4 cos (x) - 1)

- sin (x) + 4 cos (x) sin (x)

Factorizar el termino común

sin (x)

= sin (x) (- 1 + 4 cos (x))

sin (x) (4 cos (x) - 1) = 0

Resolver cada parte por separado

sin (x) = 0 or 4 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluciones generales para

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

4 cos (x) - 1 = 0 : x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

4 cos (x) - 1 = 0

Mover

1

al lado derecho

4 cos (x) - 1 = 0

Sumar

1

a ambos lados

4 cos (x) - 1 + 1 = 0 + 1

Simplificar

4 cos (x) = 1

4 cos (x) = 1

Dividir ambos lados entre

4

4 cos (x) = 1

Dividir ambos lados entre

4

\frac{4 cos ( x )}{4} = \frac{1}{4}

Simplificar

cos (x) = \frac{1}{4}

cos (x) = \frac{1}{4}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{1}{4}

Soluciones generales para

cos (x) = \frac{1}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Mostrar soluciones en forma decimal

x = 2 πn, x = π + 2 πn, x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

sin (\frac{11 π}{4})

arccos (\frac{4}{5})

cos (θ) = 1

1 + sin (θ) = 2 cos^{2} (θ)

arccsc (- 2)