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Popolare Trigonometria >

cos(4θ)=cos(2θ),180<= θ<= 360

Soluzione

cos (4 θ) = cos (2 θ), 18 0^{\circ} \leq θ \leq 36 0^{\circ}

Soluzione

θ = 18 0^{\circ}, θ = 24 0^{\circ}, θ = 30 0^{\circ}, θ = 36 0^{\circ}

Radianti

θ = π, θ = \frac{4 π}{3}, θ = \frac{5 π}{3}, θ = 2 π

Fasi della soluzione

cos (4 θ) = cos (2 θ), 18 0^{\circ} \leq θ \leq 36 0^{\circ}

Sottrarre

cos (2 θ)

da entrambi i lati

cos (4 θ) - cos (2 θ) = 0

Riscrivere utilizzando identità trigonometriche

- cos (2 θ) + cos (4 θ)

Usa la formula della somma al prodotto:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= - 2 sin (\frac{4 θ + 2 θ}{2}) sin (\frac{4 θ - 2 θ}{2})

Semplificare

- 2 sin (\frac{4 θ + 2 θ}{2}) sin (\frac{4 θ - 2 θ}{2}) : - 2 sin (3 θ) sin (θ)

- 2 sin (\frac{4 θ + 2 θ}{2}) sin (\frac{4 θ - 2 θ}{2})

\frac{4 θ + 2 θ}{2} = 3 θ

\frac{4 θ + 2 θ}{2}

Aggiungi elementi simili:

4 θ + 2 θ = 6 θ

= \frac{6 θ}{2}

Dividi i numeri:

\frac{6}{2} = 3

= 3 θ

= - 2 sin (3 θ) sin (\frac{4 θ - 2 θ}{2})

\frac{4 θ - 2 θ}{2} = θ

\frac{4 θ - 2 θ}{2}

Aggiungi elementi simili:

4 θ - 2 θ = 2 θ

= \frac{2 θ}{2}

Dividi i numeri:

\frac{2}{2} = 1

= θ

= - 2 sin (3 θ) sin (θ)

= - 2 sin (3 θ) sin (θ)

- 2 sin (3 θ) sin (θ) = 0

Risolvere ogni parte separatamente

sin (3 θ) = 0 or sin (θ) = 0

sin (3 θ) = 0, 18 0^{\circ} \leq θ \leq 36 0^{\circ} : θ = 18 0^{\circ}, θ = 24 0^{\circ}, θ = 30 0^{\circ}, θ = 36 0^{\circ}

sin (3 θ) = 0, 18 0^{\circ} \leq θ \leq 36 0^{\circ}

Soluzioni generali per

sin (3 θ) = 0

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}

3 θ = 0 + 36 0^{\circ} n, 3 θ = 18 0^{\circ} + 36 0^{\circ} n

3 θ = 0 + 36 0^{\circ} n, 3 θ = 18 0^{\circ} + 36 0^{\circ} n

Risolvi

3 θ = 0 + 36 0^{\circ} n : θ = \frac{36 0 ^{\circ} n}{3}

3 θ = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

3 θ = 36 0^{\circ} n

Dividere entrambi i lati per

3

3 θ = 36 0^{\circ} n

Dividere entrambi i lati per

3

\frac{3 θ}{3} = \frac{36 0 ^{\circ} n}{3}

Semplificare

θ = \frac{36 0 ^{\circ} n}{3}

θ = \frac{36 0 ^{\circ} n}{3}

Risolvi

3 θ = 18 0^{\circ} + 36 0^{\circ} n : θ = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

3 θ = 18 0^{\circ} + 36 0^{\circ} n

Dividere entrambi i lati per

3

3 θ = 18 0^{\circ} + 36 0^{\circ} n

Dividere entrambi i lati per

3

\frac{3 θ}{3} = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Semplificare

θ = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = \frac{36 0 ^{\circ} n}{3}, θ = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Soluzioni per l'intervallo

18 0^{\circ} \leq θ \leq 36 0^{\circ}

θ = 18 0^{\circ}, θ = 24 0^{\circ}, θ = 30 0^{\circ}, θ = 36 0^{\circ}

sin (θ) = 0, 18 0^{\circ} \leq θ \leq 36 0^{\circ} : θ = 18 0^{\circ}, θ = 36 0^{\circ}

sin (θ) = 0, 18 0^{\circ} \leq θ \leq 36 0^{\circ}

Soluzioni generali per

sin (θ) = 0

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}

θ = 0 + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

θ = 0 + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Risolvi

θ = 0 + 36 0^{\circ} n : θ = 36 0^{\circ} n

θ = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

θ = 36 0^{\circ} n

θ = 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Soluzioni per l'intervallo

18 0^{\circ} \leq θ \leq 36 0^{\circ}

θ = 18 0^{\circ}, θ = 36 0^{\circ}

Combinare tutte le soluzioni

θ = 18 0^{\circ}, θ = 24 0^{\circ}, θ = 30 0^{\circ}, θ = 36 0^{\circ}

Grafico

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