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Popolare >

sin(4θ)=(sqrt(3))/2 ,0<= θ<2pi

Soluzione

sin (4 θ) = \frac{3}{2}, 0 \leq θ < 2 π

Soluzione

θ = \frac{π}{12}, θ = \frac{π}{6}, θ = \frac{7 π}{12}, θ = \frac{2 π}{3}, θ = \frac{13 π}{12}, θ = \frac{7 π}{6}, θ = \frac{19 π}{12}, θ = \frac{5 π}{3}

Gradi

θ = 1 5^{\circ}, θ = 3 0^{\circ}, θ = 10 5^{\circ}, θ = 12 0^{\circ}, θ = 19 5^{\circ}, θ = 21 0^{\circ}, θ = 28 5^{\circ}, θ = 30 0^{\circ}

Fasi della soluzione

sin (4 θ) = \frac{3}{2}, 0 \leq θ < 2 π

Soluzioni generali per

sin (4 θ) = \frac{3}{2}

sin (x)

periodicità tabella con

2 πn

cicli:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

4 θ = \frac{π}{3} + 2 πn, 4 θ = \frac{2 π}{3} + 2 πn

4 θ = \frac{π}{3} + 2 πn, 4 θ = \frac{2 π}{3} + 2 πn

Risolvi

4 θ = \frac{π}{3} + 2 πn : θ = \frac{π}{12} + \frac{πn}{2}

4 θ = \frac{π}{3} + 2 πn

Dividere entrambi i lati per

4

4 θ = \frac{π}{3} + 2 πn

Dividere entrambi i lati per

4

\frac{4 θ}{4} = \frac{\frac{π}{3}}{4} + \frac{2 πn}{4}

Semplificare

\frac{4 θ}{4} = \frac{\frac{π}{3}}{4} + \frac{2 πn}{4}

Semplificare

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Dividi i numeri:

\frac{4}{4} = 1

= θ

Semplificare

\frac{\frac{π}{3}}{4} + \frac{2 πn}{4} : \frac{π}{12} + \frac{πn}{2}

\frac{\frac{π}{3}}{4} + \frac{2 πn}{4}

\frac{\frac{π}{3}}{4} = \frac{π}{12}

\frac{\frac{π}{3}}{4}

Applica la regola delle frazioni:

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

= \frac{π}{3 \cdot 4}

Moltiplica i numeri:

3 \cdot 4 = 12

= \frac{π}{12}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancella il fattore comune:

2

= \frac{πn}{2}

= \frac{π}{12} + \frac{πn}{2}

θ = \frac{π}{12} + \frac{πn}{2}

θ = \frac{π}{12} + \frac{πn}{2}

θ = \frac{π}{12} + \frac{πn}{2}

Risolvi

4 θ = \frac{2 π}{3} + 2 πn : θ = \frac{π}{6} + \frac{πn}{2}

4 θ = \frac{2 π}{3} + 2 πn

Dividere entrambi i lati per

4

4 θ = \frac{2 π}{3} + 2 πn

Dividere entrambi i lati per

4

\frac{4 θ}{4} = \frac{\frac{2 π}{3}}{4} + \frac{2 πn}{4}

Semplificare

\frac{4 θ}{4} = \frac{\frac{2 π}{3}}{4} + \frac{2 πn}{4}

Semplificare

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Dividi i numeri:

\frac{4}{4} = 1

= θ

Semplificare

\frac{\frac{2 π}{3}}{4} + \frac{2 πn}{4} : \frac{π}{6} + \frac{πn}{2}

\frac{\frac{2 π}{3}}{4} + \frac{2 πn}{4}

\frac{\frac{2 π}{3}}{4} = \frac{π}{6}

\frac{\frac{2 π}{3}}{4}

Applica la regola delle frazioni:

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

= \frac{2 π}{3 \cdot 4}

Moltiplica i numeri:

3 \cdot 4 = 12

= \frac{2 π}{12}

Cancella il fattore comune:

2

= \frac{π}{6}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancella il fattore comune:

2

= \frac{πn}{2}

= \frac{π}{6} + \frac{πn}{2}

θ = \frac{π}{6} + \frac{πn}{2}

θ = \frac{π}{6} + \frac{πn}{2}

θ = \frac{π}{6} + \frac{πn}{2}

θ = \frac{π}{12} + \frac{πn}{2}, θ = \frac{π}{6} + \frac{πn}{2}

Soluzioni per l'intervallo

0 \leq θ < 2 π

θ = \frac{π}{12}, θ = \frac{π}{6}, θ = \frac{7 π}{12}, θ = \frac{2 π}{3}, θ = \frac{13 π}{12}, θ = \frac{7 π}{6}, θ = \frac{19 π}{12}, θ = \frac{5 π}{3}

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