3sin(x)+6cos(x)=4

Solución

3 sin (x) + 6 cos (x) = 4

x = 1.39557 \dots + 2 πn, x = 2 π - 0.46828 \dots + 2 πn

Pasos de solución

3 sin (x) + 6 cos (x) = 4

Restar

6 cos (x)

de ambos lados

3 sin (x) = 4 - 6 cos (x)

Elevar al cuadrado ambos lados

(3 sin (x))^{2} = (4 - 6 cos (x))^{2}

Restar

(4 - 6 cos (x))^{2}

de ambos lados

9 sin^{2} (x) - 16 + 48 cos (x) - 36 cos^{2} (x) = 0

Re-escribir usando identidades trigonométricas

- 7 - 45 cos^{2} (x) + 48 cos (x) = 0

Usando el método de sustitución

cos (x) = \frac{8 - 29}{15}, cos (x) = \frac{8 + 29}{15}

cos (x) = \frac{8 - 29}{15} : x = arccos (\frac{8 - 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 - 29}{15}) + 2 πn

cos (x) = \frac{8 + 29}{15} : x = arccos (\frac{8 + 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 + 29}{15}) + 2 πn

Combinar toda las soluciones

x = arccos (\frac{8 - 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 - 29}{15}) + 2 πn, x = arccos (\frac{8 + 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 + 29}{15}) + 2 πn

Verificar las soluciones sustituyendo en la ecuación original

x = arccos (\frac{8 - 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 + 29}{15}) + 2 πn

Mostrar soluciones en forma decimal

x = 1.39557 \dots + 2 πn, x = 2 π - 0.46828 \dots + 2 πn

3 tan (2 x + 1) = 2

\frac{1}{( 1 + tan ( a ) )} = cot (a)

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

sin (x - \frac{π}{6}) = 0

2 = \frac{1}{cos ^{2} ( x )}