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sin(x+60)=2cos(x)

Solución

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

Solución

x = 1.15551 \dots + 18 0^{\circ} n

radianes

x = 1.15551 \dots + πn

Pasos de solución

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

Re-escribir usando identidades trigonométricas

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad de suma 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)

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

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

Restar

2 cos (x)

de ambos lados

\frac{1}{2} sin (x) + \frac{3 - 4}{2} cos (x) = 0

Simplificar

\frac{1}{2} sin (x) + \frac{3 - 4}{2} cos (x) : \frac{sin ( x ) + ( 3 - 4 ) cos ( x )}{2}

\frac{1}{2} sin (x) + \frac{3 - 4}{2} 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 - 4}{2} cos (x) = \frac{( 3 - 4 ) cos ( x )}{2}

\frac{3 - 4}{2} cos (x)

Multiplicar fracciones:

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

= \frac{( 3 - 4 ) cos ( x )}{2}

= \frac{sin ( x )}{2} + \frac{( 3 - 4 ) cos ( x )}{2}

Aplicar la regla

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

= \frac{sin ( x ) + ( 3 - 4 ) cos ( x )}{2}

\frac{sin ( x ) + ( 3 - 4 ) cos ( x )}{2} = 0

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

sin (x) + (3 - 4) cos (x) = 0

Re-escribir usando identidades trigonométricas

sin (x) + (3 - 4) cos (x) = 0

Dividir ambos lados entre

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

\frac{sin ( x ) + ( 3 - 4 ) cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplificar

\frac{sin ( x )}{cos ( x )} + 3 - 4 = 0

Utilizar la identidad trigonométrica básica:

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

tan (x) + 3 - 4 = 0

tan (x) + 3 - 4 = 0

Mover

3

al lado derecho

tan (x) + 3 - 4 = 0

Restar

3

de ambos lados

tan (x) + 3 - 4 - 3 = 0 - 3

Simplificar

tan (x) - 4 = - 3

tan (x) - 4 = - 3

Mover

4

al lado derecho

tan (x) - 4 = - 3

Sumar

4

a ambos lados

tan (x) - 4 + 4 = - 3 + 4

Simplificar

tan (x) = - 3 + 4

tan (x) = - 3 + 4

Aplicar propiedades trigonométricas inversas

tan (x) = - 3 + 4

Soluciones generales para

tan (x) = - 3 + 4

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

x = arctan (- 3 + 4) + 18 0^{\circ} n

x = arctan (- 3 + 4) + 18 0^{\circ} n

Mostrar soluciones en forma decimal

x = 1.15551 \dots + 18 0^{\circ} n

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sin (9 x) = sin (x)

cot^3(x)+2cot^2(x)-3cot(x)-6=0

cot^{3} (x) + 2 cot^{2} (x) - 3 cot (x) - 6 = 0

sin^2(x)=cos^3(x)

sin^{2} (x) = cos^{3} (x)

12cot^2(x)-cot(x)=1

12 cot^{2} (x) - cot (x) = 1

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