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Popular >

verificar sin(x+30)+sqrt(3)cos(x+30)=2cos(x)

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Manipular el lado derecho

sin (x + 3 0^{\circ}) + 3 cos (x + 3 0^{\circ})

Re-escribir usando identidades trigonométricas

sin (x + 3 0^{\circ})

Utilizar la identidad de suma de ángulos:

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

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

Simplificar

sin (x) cos (3 0^{\circ}) + cos (x) sin (3 0^{\circ}) : \frac{3}{2} sin (x) + \frac{1}{2} cos (x)

sin (x) cos (3 0^{\circ}) + cos (x) sin (3 0^{\circ})

Simplificar

cos (3 0^{\circ}) : \frac{3}{2}

cos (3 0^{\circ})

Utilizar la siguiente identidad trivial:

cos (3 0^{\circ}) = \frac{3}{2}

cos (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

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

Simplificar

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Utilizar la siguiente identidad trivial:

sin (3 0^{\circ}) = \frac{1}{2}

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{3}{2} sin (x) + \frac{1}{2} cos (x)

= \frac{3}{2} sin (x) + \frac{1}{2} cos (x)

= \frac{3}{2} sin (x) + \frac{1}{2} cos (x) + 3 cos (x + 3 0^{\circ})

Re-escribir usando identidades trigonométricas

cos (x + 3 0^{\circ})

Utilizar la identidad de suma de ángulos:

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

= cos (x) cos (3 0^{\circ}) - sin (x) sin (3 0^{\circ})

Simplificar

cos (x) cos (3 0^{\circ}) - sin (x) sin (3 0^{\circ}) : \frac{3}{2} cos (x) - \frac{1}{2} sin (x)

cos (x) cos (3 0^{\circ}) - sin (x) sin (3 0^{\circ})

Simplificar

cos (3 0^{\circ}) : \frac{3}{2}

cos (3 0^{\circ})

Utilizar la siguiente identidad trivial:

cos (3 0^{\circ}) = \frac{3}{2}

cos (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

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

Simplificar

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Utilizar la siguiente identidad trivial:

sin (3 0^{\circ}) = \frac{1}{2}

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{3}{2} cos (x) - \frac{1}{2} sin (x)

= \frac{3}{2} cos (x) - \frac{1}{2} sin (x)

= \frac{3}{2} sin (x) + \frac{1}{2} cos (x) + 3 (\frac{3}{2} cos (x) - \frac{1}{2} sin (x))

Simplificar

\frac{3}{2} sin (x) + \frac{1}{2} cos (x) + 3 (\frac{3}{2} cos (x) - \frac{1}{2} sin (x)) : 2 cos (x)

\frac{3}{2} sin (x) + \frac{1}{2} cos (x) + 3 (\frac{3}{2} cos (x) - \frac{1}{2} sin (x))

Expandir

3 (\frac{3}{2} cos (x) - \frac{1}{2} sin (x)) : \frac{3}{2} cos (x) - \frac{3}{2} sin (x)

3 (\frac{3}{2} cos (x) - \frac{1}{2} sin (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 3, b = \frac{3}{2} cos (x), c = \frac{1}{2} sin (x)

= 3 \frac{3}{2} cos (x) - 3 \frac{1}{2} sin (x)

Simplificar

3 \frac{3}{2} cos (x) - 3 \frac{1}{2} sin (x) : \frac{3}{2} cos (x) - \frac{3}{2} sin (x)

3 \frac{3}{2} cos (x) - 3 \frac{1}{2} sin (x)

3 \frac{3}{2} cos (x) = \frac{3}{2} cos (x)

3 \frac{3}{2} cos (x)

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 3}{2} cos (x)

33 = 3

33

Aplicar las leyes de los exponentes:

a a = a

33 = 3

= 3

= \frac{3}{2} cos (x)

3 \frac{1}{2} sin (x) = \frac{3}{2} sin (x)

3 \frac{1}{2} sin (x)

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 3}{2} sin (x)

Multiplicar:

1 \cdot 3 = 3

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

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

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

= \frac{3}{2} sin (x) + \frac{1}{2} cos (x) + \frac{3}{2} cos (x) - \frac{3}{2} sin (x)

Simplificar

\frac{3}{2} sin (x) + \frac{1}{2} cos (x) + \frac{3}{2} cos (x) - \frac{3}{2} sin (x) : 2 cos (x)

\frac{3}{2} sin (x) + \frac{1}{2} cos (x) + \frac{3}{2} cos (x) - \frac{3}{2} sin (x)

Agrupar términos semejantes

= \frac{1}{2} cos (x) + \frac{3}{2} cos (x) + \frac{3}{2} sin (x) - \frac{3}{2} sin (x)

Sumar elementos similares:

\frac{1}{2} cos (x) + \frac{3}{2} cos (x) = 2 cos (x)

\frac{1}{2} cos (x) + \frac{3}{2} cos (x)

Factorizar el termino común

cos (x)

= cos (x) (\frac{1}{2} + \frac{3}{2})

\frac{1}{2} + \frac{3}{2} = 2

\frac{1}{2} + \frac{3}{2}

Aplicar la regla

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 3}{2}

Sumar:

1 + 3 = 4

= \frac{4}{2}

Dividir:

\frac{4}{2} = 2

= 2

= 2 cos (x)

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

Sumar elementos similares:

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

\frac{3}{2} sin (x) - \frac{3}{2} sin (x)

Factorizar el termino común

sin (x)

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

\frac{3}{2} - \frac{3}{2} = 0

\frac{3}{2} - \frac{3}{2}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 - 3}{2}

Factorizar

3 - 3 : 0

3 - 3

Factorizar el termino común

3

= 3 (1 - 1)

Simplificar

= 0

= \frac{0}{2}

Aplicar la regla

\frac{0}{a} = 0, a \neq = 0

= 0

= 0

= 2 cos (x)

= 2 cos (x)

= 2 cos (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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