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2sin(x-60)=cos(x-30)

Solución

2 sin (x - 6 0^{\circ}) = cos (x - 3 0^{\circ})

Solución

x = 1.38067 \dots + 18 0^{\circ} n

radianes

x = 1.38067 \dots + πn

Pasos de solución

2 sin (x - 6 0^{\circ}) = cos (x - 3 0^{\circ})

Re-escribir usando identidades trigonométricas

2 sin (x - 6 0^{\circ}) = cos (x - 3 0^{\circ})

Re-escribir usando identidades trigonométricas

sin (x - 6 0^{\circ})

Utilizar la identidad de diferencia de ángulos:

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

= sin (x) cos (6 0^{\circ}) - cos (x) sin (6 0^{\circ})

Simplificar

sin (x) cos (6 0^{\circ}) - cos (x) sin (6 0^{\circ}) : \frac{1}{2} sin (x) - \frac{3}{2} cos (x)

sin (x) cos (6 0^{\circ}) - cos (x) sin (6 0^{\circ})

Simplificar

cos (6 0^{\circ}) : \frac{1}{2}

cos (6 0^{\circ})

Utilizar la siguiente identidad trivial:

cos (6 0^{\circ}) = \frac{1}{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{1}{2}

= \frac{1}{2} sin (x) - sin (6 0^{\circ}) cos (x)

Simplificar

sin (6 0^{\circ}) : \frac{3}{2}

sin (6 0^{\circ})

Utilizar la siguiente identidad trivial:

sin (6 0^{\circ}) = \frac{3}{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{3}{2}

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

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

Utilizar la identidad de diferencia 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)

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

Simplificar

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

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

Simplificar

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

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

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= sin (x) \cdot 1

Multiplicar:

sin (x) \cdot 1 = sin (x)

= sin (x)

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= cos (x) 3

= sin (x) - 3 cos (x)

= sin (x) - 3 cos (x)

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

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

Restar

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

de ambos lados

\frac{1}{2} sin (x) - \frac{3}{2} cos (x) - 3 cos (x) = 0

Simplificar

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

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

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

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

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

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

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

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

Multiplicar fracciones:

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

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

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

Combinar las fracciones usando el mínimo común denominador:

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

Aplicar la regla

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

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

Sumar elementos similares:

- 3 cos (x) - 23 cos (x) = - 33 cos (x)

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

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

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

sin (x) - 33 cos (x) = 0

Re-escribir usando identidades trigonométricas

sin (x) - 33 cos (x) = 0

Dividir ambos lados entre

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

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

Simplificar

\frac{sin ( x )}{cos ( x )} - 33 = 0

Utilizar la identidad trigonométrica básica:

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

tan (x) - 33 = 0

tan (x) - 33 = 0

Mover

33

al lado derecho

tan (x) - 33 = 0

Sumar

33

a ambos lados

tan (x) - 33 + 33 = 0 + 33

Simplificar

tan (x) = 33

tan (x) = 33

Aplicar propiedades trigonométricas inversas

tan (x) = 33

Soluciones generales para

tan (x) = 33

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

x = arctan (33) + 18 0^{\circ} n

x = arctan (33) + 18 0^{\circ} n

Mostrar soluciones en forma decimal

x = 1.38067 \dots + 18 0^{\circ} n

Ejemplos populares

2sin(θ)=2cos(θ)

2 sin (θ) = 2 cos (θ)

2=2cos(3θ)

2 = 2 cos (3 θ)

sin(x)=sin(pi/2-x)

sin (x) = sin (\frac{π}{2} - x)

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

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

cot(x/2)=sin(x)

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