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Popolare Trigonometria >

120sin(x-120)=120sin(x-240)

Soluzione

120 sin (x - 12 0^{\circ}) = 120 sin (x - 24 0^{\circ})

Soluzione

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Radianti

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Fasi della soluzione

120 sin (x - 12 0^{\circ}) = 120 sin (x - 24 0^{\circ})

Riscrivere utilizzando identità trigonometriche

120 sin (x - 12 0^{\circ}) = 120 sin (x - 24 0^{\circ})

Riscrivere utilizzando identità trigonometriche

sin (x - 12 0^{\circ})

Usa la formula della differenza degli angoli:

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

= sin (x) cos (12 0^{\circ}) - cos (x) sin (12 0^{\circ})

Semplifica

sin (x) cos (12 0^{\circ}) - cos (x) sin (12 0^{\circ}) : - \frac{1}{2} sin (x) - \frac{3}{2} cos (x)

sin (x) cos (12 0^{\circ}) - cos (x) sin (12 0^{\circ})

sin (x) cos (12 0^{\circ}) = - \frac{1}{2} sin (x)

sin (x) cos (12 0^{\circ})

Semplifica

cos (12 0^{\circ}) : - \frac{1}{2}

cos (12 0^{\circ})

Usare la seguente identità triviale:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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)

Rimuovi le parentesi:

(- a) = - a

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

= - \frac{1}{2} sin (x) - sin (12 0^{\circ}) cos (x)

Semplifica

sin (12 0^{\circ}) : \frac{3}{2}

sin (12 0^{\circ})

Usare la seguente identità triviale:

sin (12 0^{\circ}) = \frac{3}{2}

sin (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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)

Usa la formula della differenza degli angoli:

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

= sin (x) cos (24 0^{\circ}) - cos (x) sin (24 0^{\circ})

Semplifica

sin (x) cos (24 0^{\circ}) - cos (x) sin (24 0^{\circ}) : - \frac{1}{2} sin (x) + \frac{3}{2} cos (x)

sin (x) cos (24 0^{\circ}) - cos (x) sin (24 0^{\circ})

sin (x) cos (24 0^{\circ}) = - \frac{1}{2} sin (x)

sin (x) cos (24 0^{\circ})

cos (24 0^{\circ}) = - \frac{1}{2}

cos (24 0^{\circ})

Riscrivere utilizzando identità trigonometriche:

cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

cos (24 0^{\circ})

Scrivere

cos (24 0^{\circ})

come

cos (18 0^{\circ} + 6 0^{\circ})

= cos (18 0^{\circ} + 6 0^{\circ})

Usa la formula della somma degli angoli:

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

= cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

= cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

Usare la seguente identità triviale:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

cos (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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}

= (- 1)

Usare la seguente identità triviale:

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

cos (6 0^{\circ})

cos (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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}

Usare la seguente identità triviale:

sin (18 0^{\circ}) = 0

sin (18 0^{\circ})

sin (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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}

= 0

Usare la seguente identità triviale:

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

sin (6 0^{\circ})

sin (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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}

= (- 1) \frac{1}{2} - 0 \cdot \frac{3}{2}

Semplificare

= - \frac{1}{2}

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

Rimuovi le parentesi:

(- a) = - a

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

= - \frac{1}{2} sin (x) - sin (24 0^{\circ}) cos (x)

sin (24 0^{\circ}) = - \frac{3}{2}

sin (24 0^{\circ})

Riscrivere utilizzando identità trigonometriche:

sin (18 0^{\circ}) cos (6 0^{\circ}) + cos (18 0^{\circ}) sin (6 0^{\circ})

sin (24 0^{\circ})

Scrivere

sin (24 0^{\circ})

come

sin (18 0^{\circ} + 6 0^{\circ})

= sin (18 0^{\circ} + 6 0^{\circ})

Usa la formula della somma degli angoli:

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

= sin (18 0^{\circ}) cos (6 0^{\circ}) + cos (18 0^{\circ}) sin (6 0^{\circ})

= sin (18 0^{\circ}) cos (6 0^{\circ}) + cos (18 0^{\circ}) sin (6 0^{\circ})

Usare la seguente identità triviale:

sin (18 0^{\circ}) = 0

sin (18 0^{\circ})

sin (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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}

= 0

Usare la seguente identità triviale:

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

cos (6 0^{\circ})

cos (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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}

Usare la seguente identità triviale:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

cos (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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}

= (- 1)

Usare la seguente identità triviale:

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

sin (6 0^{\circ})

sin (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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}

= 0 \cdot \frac{1}{2} + (- 1) \frac{3}{2}

Semplificare

= - \frac{3}{2}

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

Applicare la regola

- (- a) = a

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

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

120 (- \frac{1}{2} sin (x) - \frac{3}{2} cos (x)) = 120 (- \frac{1}{2} sin (x) + \frac{3}{2} cos (x))

120 (- \frac{1}{2} sin (x) - \frac{3}{2} cos (x)) = 120 (- \frac{1}{2} sin (x) + \frac{3}{2} cos (x))

Sottrarre

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

da entrambi i lati

- 1203 cos (x) = 0

Dividere entrambi i lati per

- 1203

- 1203 cos (x) = 0

Dividere entrambi i lati per

- 1203

\frac{- 120 3 cos ( x )}{- 120 3} = \frac{0}{- 120 3}

Semplificare

cos (x) = 0

cos (x) = 0

Soluzioni generali per

cos (x) = 0

cos (x)

periodicità tabella con

36 0^{\circ} n

cicli:

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}

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Grafico

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