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Popular Trigonometría >

4sin(2x-0.4)-5cos(2x-0.4)=0

Solución

4 sin (2 x - 0.4) - 5 cos (2 x - 0.4) = 0

Solución

x = \frac{1.29605 \dots}{2} + \frac{πn}{2}

Grados

x = 37.12925 \dots^{\circ} + 9 0^{\circ} n

Pasos de solución

4 sin (2 x - 0.4) - 5 cos (2 x - 0.4) = 0

Re-escribir usando identidades trigonométricas

4 sin (2 x - 0.4) - 5 cos (2 x - 0.4) = 0

Re-escribir usando identidades trigonométricas

sin (2 x - 0.4)

Utilizar la identidad de diferencia de ángulos:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (2 x) cos (0.4) - cos (2 x) sin (0.4)

Simplificar

sin (2 x) cos (0.4) - cos (2 x) sin (0.4) : 0.92106 \dots sin (2 x) - 0.38941 \dots cos (2 x)

sin (2 x) cos (0.4) - cos (2 x) sin (0.4)

Simplificar

cos (0.4) : 0.92106 \dots

cos (0.4)

cos (0.4) = 0.92106 \dots

= 0.92106 \dots

= 0.92106 \dots sin (2 x) - sin (0.4) cos (2 x)

Simplificar

sin (0.4) : 0.38941 \dots

sin (0.4)

sin (0.4) = 0.38941 \dots

= 0.38941 \dots

= 0.92106 \dots sin (2 x) - 0.38941 \dots cos (2 x)

= 0.92106 \dots sin (2 x) - 0.38941 \dots cos (2 x)

Utilizar la identidad de diferencia de ángulos:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= cos (2 x) cos (0.4) + sin (2 x) sin (0.4)

Simplificar

cos (2 x) cos (0.4) + sin (2 x) sin (0.4) : 0.92106 \dots cos (2 x) + 0.38941 \dots sin (2 x)

cos (2 x) cos (0.4) + sin (2 x) sin (0.4)

Simplificar

cos (0.4) : 0.92106 \dots

cos (0.4)

cos (0.4) = 0.92106 \dots

= 0.92106 \dots

= 0.92106 \dots cos (2 x) + sin (0.4) sin (2 x)

Simplificar

sin (0.4) : 0.38941 \dots

sin (0.4)

sin (0.4) = 0.38941 \dots

= 0.38941 \dots

= 0.92106 \dots cos (2 x) + 0.38941 \dots sin (2 x)

= 0.92106 \dots cos (2 x) + 0.38941 \dots sin (2 x)

4 (0.92106 \dots sin (2 x) - 0.38941 \dots cos (2 x)) - 5 (0.92106 \dots cos (2 x) + 0.38941 \dots sin (2 x)) = 0

Simplificar

4 (0.92106 \dots sin (2 x) - 0.38941 \dots cos (2 x)) - 5 (0.92106 \dots cos (2 x) + 0.38941 \dots sin (2 x)) : 1.73715 \dots sin (2 x) - 6.16297 \dots cos (2 x)

4 (0.92106 \dots sin (2 x) - 0.38941 \dots cos (2 x)) - 5 (0.92106 \dots cos (2 x) + 0.38941 \dots sin (2 x))

Expandir

4 (0.92106 \dots sin (2 x) - 0.38941 \dots cos (2 x)) : 3.68424 \dots sin (2 x) - 1.55767 \dots cos (2 x)

4 (0.92106 \dots sin (2 x) - 0.38941 \dots cos (2 x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 4, b = 0.92106 \dots sin (2 x), c = 0.38941 \dots cos (2 x)

= 4 \cdot 0.92106 \dots sin (2 x) - 4 \cdot 0.38941 \dots cos (2 x)

Simplificar

4 \cdot 0.92106 \dots sin (2 x) - 4 \cdot 0.38941 \dots cos (2 x) : 3.68424 \dots sin (2 x) - 1.55767 \dots cos (2 x)

4 \cdot 0.92106 \dots sin (2 x) - 4 \cdot 0.38941 \dots cos (2 x)

Multiplicar los numeros:

4 \cdot 0.92106 \dots = 3.68424 \dots

= 3.68424 \dots sin (2 x) - 4 \cdot 0.38941 \dots cos (2 x)

Multiplicar los numeros:

4 \cdot 0.38941 \dots = 1.55767 \dots

= 3.68424 \dots sin (2 x) - 1.55767 \dots cos (2 x)

= 3.68424 \dots sin (2 x) - 1.55767 \dots cos (2 x)

= 3.68424 \dots sin (2 x) - 1.55767 \dots cos (2 x) - 5 (0.92106 \dots cos (2 x) + 0.38941 \dots sin (2 x))

Expandir

- 5 (0.92106 \dots cos (2 x) + 0.38941 \dots sin (2 x)) : - 4.60530 \dots cos (2 x) - 1.94709 \dots sin (2 x)

- 5 (0.92106 \dots cos (2 x) + 0.38941 \dots sin (2 x))

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = - 5, b = 0.92106 \dots cos (2 x), c = 0.38941 \dots sin (2 x)

= - 5 \cdot 0.92106 \dots cos (2 x) + (- 5) \cdot 0.38941 \dots sin (2 x)

Aplicar las reglas de los signos

+ (- a) = - a

= - 5 \cdot 0.92106 \dots cos (2 x) - 5 \cdot 0.38941 \dots sin (2 x)

Simplificar

- 5 \cdot 0.92106 \dots cos (2 x) - 5 \cdot 0.38941 \dots sin (2 x) : - 4.60530 \dots cos (2 x) - 1.94709 \dots sin (2 x)

- 5 \cdot 0.92106 \dots cos (2 x) - 5 \cdot 0.38941 \dots sin (2 x)

Multiplicar los numeros:

5 \cdot 0.92106 \dots = 4.60530 \dots

= - 4.60530 \dots cos (2 x) - 5 \cdot 0.38941 \dots sin (2 x)

Multiplicar los numeros:

5 \cdot 0.38941 \dots = 1.94709 \dots

= - 4.60530 \dots cos (2 x) - 1.94709 \dots sin (2 x)

= - 4.60530 \dots cos (2 x) - 1.94709 \dots sin (2 x)

= 3.68424 \dots sin (2 x) - 1.55767 \dots cos (2 x) - 4.60530 \dots cos (2 x) - 1.94709 \dots sin (2 x)

Simplificar

3.68424 \dots sin (2 x) - 1.55767 \dots cos (2 x) - 4.60530 \dots cos (2 x) - 1.94709 \dots sin (2 x) : 1.73715 \dots sin (2 x) - 6.16297 \dots cos (2 x)

3.68424 \dots sin (2 x) - 1.55767 \dots cos (2 x) - 4.60530 \dots cos (2 x) - 1.94709 \dots sin (2 x)

Sumar elementos similares:

- 1.55767 \dots cos (2 x) - 4.60530 \dots cos (2 x) = - 6.16297 \dots cos (2 x)

= 3.68424 \dots sin (2 x) - 6.16297 \dots cos (2 x) - 1.94709 \dots sin (2 x)

Sumar elementos similares:

3.68424 \dots sin (2 x) - 1.94709 \dots sin (2 x) = 1.73715 \dots sin (2 x)

= 1.73715 \dots sin (2 x) - 6.16297 \dots cos (2 x)

= 1.73715 \dots sin (2 x) - 6.16297 \dots cos (2 x)

1.73715 \dots sin (2 x) - 6.16297 \dots cos (2 x) = 0

Dividir ambos lados entre

cos (2 x), cos (2 x) \neq = 0

\frac{1.73715 \dots sin ( 2 x ) - 6.16297 \dots cos ( 2 x )}{cos ( 2 x )} = \frac{0}{cos ( 2 x )}

Simplificar

\frac{1.73715 \dots sin ( 2 x )}{cos ( 2 x )} - 6.16297 \dots = 0

Utilizar la identidad trigonométrica básica:

\frac{sin ( x )}{cos ( x )} = tan (x)

1.73715 \dots tan (2 x) - 6.16297 \dots = 0

1.73715 \dots tan (2 x) - 6.16297 \dots = 0

Desplace

6.16297 \dots

a la derecha

1.73715 \dots tan (2 x) - 6.16297 \dots = 0

Sumar

6.16297 \dots

a ambos lados

1.73715 \dots tan (2 x) - 6.16297 \dots + 6.16297 \dots = 0 + 6.16297 \dots

Simplificar

1.73715 \dots tan (2 x) = 6.16297 \dots

1.73715 \dots tan (2 x) = 6.16297 \dots

Dividir ambos lados entre

1.73715 \dots

1.73715 \dots tan (2 x) = 6.16297 \dots

Dividir ambos lados entre

1.73715 \dots

\frac{1.73715 \dots tan ( 2 x )}{1.73715 \dots} = \frac{6.16297 \dots}{1.73715 \dots}

Simplificar

tan (2 x) = 3.54774 \dots

tan (2 x) = 3.54774 \dots

Aplicar propiedades trigonométricas inversas

tan (2 x) = 3.54774 \dots

Soluciones generales para

tan (2 x) = 3.54774 \dots

tan (x) = a \Rightarrow x = arctan (a) + πn

2 x = arctan (3.54774 \dots) + πn

2 x = arctan (3.54774 \dots) + πn

Resolver

2 x = arctan (3.54774 \dots) + πn : x = \frac{arctan ( 3.54774 \dots )}{2} + \frac{πn}{2}

2 x = arctan (3.54774 \dots) + πn

Dividir ambos lados entre

2

2 x = arctan (3.54774 \dots) + πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{arctan ( 3.54774 \dots )}{2} + \frac{πn}{2}

Simplificar

x = \frac{arctan ( 3.54774 \dots )}{2} + \frac{πn}{2}

x = \frac{arctan ( 3.54774 \dots )}{2} + \frac{πn}{2}

x = \frac{arctan ( 3.54774 \dots )}{2} + \frac{πn}{2}

Mostrar soluciones en forma decimal

x = \frac{1.29605 \dots}{2} + \frac{πn}{2}

Gráfica

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