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Popular Trigonometria >

provar sin^3(θ)= 3/4 sin(θ)-1/4 sin(3θ)

Solução

provar

sin^{3} (θ) = \frac{3}{4} sin (θ) - \frac{1}{4} sin (3 θ)

Solução

Verdadeiro

Passos da solução

sin^{3} (θ) = \frac{3}{4} sin (θ) - \frac{1}{4} sin (3 θ)

Manipular o lado esquerdo

\frac{3}{4} sin (θ) - \frac{1}{4} sin (3 θ)

Reeecreva usando identidades trigonométricas

\frac{3}{4} sin (θ) - \frac{1}{4} sin (3 θ)

Usar a seguinte identidade:

sin (3 x) = 3 sin (x) - 4 sin^{3} (x)

sin (3 x)

Reeecreva usando identidades trigonométricas

sin (3 x)

Reescrever como

= sin (2 x + x)

Use a identidade de soma de ângulos:

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

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

Utilizar a identidade trigonométrica do arco duplo:

sin (2 x) = 2 sin (x) cos (x)

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

Simplificar

cos (2 x) sin (x) + cos (x) \cdot 2 sin (x) cos (x) : sin (x) cos (2 x) + 2 cos^{2} (x) sin (x)

cos (2 x) sin (x) + cos (x) 2 sin (x) cos (x)

cos (x) \cdot 2 sin (x) cos (x) = 2 cos^{2} (x) sin (x)

cos (x) 2 sin (x) cos (x)

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= 2 sin (x) cos^{1 + 1} (x)

Somar:

1 + 1 = 2

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

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

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

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

Utilizar a identidade trigonométrica do arco duplo:

cos (2 x) = 1 - 2 sin^{2} (x)

= (1 - 2 sin^{2} (x)) sin (x) + 2 cos^{2} (x) sin (x)

Utilizar a identidade trigonométrica pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= (1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x)

Expandir

(1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x) : - 4 sin^{3} (x) + 3 sin (x)

(1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x)

= sin (x) (1 - 2 sin^{2} (x)) + 2 sin (x) (1 - sin^{2} (x))

Expandir

sin (x) (1 - 2 sin^{2} (x)) : sin (x) - 2 sin^{3} (x)

sin (x) (1 - 2 sin^{2} (x))

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = sin (x), b = 1, c = 2 sin^{2} (x)

= sin (x) 1 - sin (x) 2 sin^{2} (x)

= 1 sin (x) - 2 sin^{2} (x) sin (x)

Simplificar

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

1 sin (x) - 2 sin^{2} (x) sin (x)

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

1 sin (x)

Multiplicar:

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

= sin (x)

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

2 sin^{2} (x) sin (x)

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (x) sin (x) = sin^{2 + 1} (x)

= 2 sin^{2 + 1} (x)

Somar:

2 + 1 = 3

= 2 sin^{3} (x)

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

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

= sin (x) - 2 sin^{3} (x) + 2 (1 - sin^{2} (x)) sin (x)

Expandir

2 sin (x) (1 - sin^{2} (x)) : 2 sin (x) - 2 sin^{3} (x)

2 sin (x) (1 - sin^{2} (x))

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = 2 sin (x), b = 1, c = sin^{2} (x)

= 2 sin (x) 1 - 2 sin (x) sin^{2} (x)

= 2 \cdot 1 sin (x) - 2 sin^{2} (x) sin (x)

Simplificar

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

2 \cdot 1 sin (x) - 2 sin^{2} (x) sin (x)

2 \cdot 1 \cdot sin (x) = 2 sin (x)

2 \cdot 1 sin (x)

Multiplicar os números:

2 \cdot 1 = 2

= 2 sin (x)

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

2 sin^{2} (x) sin (x)

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (x) sin (x) = sin^{2 + 1} (x)

= 2 sin^{2 + 1} (x)

Somar:

2 + 1 = 3

= 2 sin^{3} (x)

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

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

= sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x)

Simplificar

sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x) : - 4 sin^{3} (x) + 3 sin (x)

sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x)

Agrupar termos semelhantes

= - 2 sin^{3} (x) - 2 sin^{3} (x) + sin (x) + 2 sin (x)

Somar elementos similares:

- 2 sin^{3} (x) - 2 sin^{3} (x) = - 4 sin^{3} (x)

= - 4 sin^{3} (x) + sin (x) + 2 sin (x)

Somar elementos similares:

sin (x) + 2 sin (x) = 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= 3 sin (x) - 4 sin^{3} (x)

= \frac{3}{4} sin (θ) - \frac{1}{4} (3 sin (θ) - 4 sin^{3} (θ))

Simplificar

\frac{3}{4} sin (θ) - \frac{1}{4} (3 sin (θ) - 4 sin^{3} (θ)) : sin^{3} (θ)

\frac{3}{4} sin (θ) - \frac{1}{4} (3 sin (θ) - 4 sin^{3} (θ))

Expandir

- \frac{1}{4} (3 sin (θ) - 4 sin^{3} (θ)) : - \frac{3}{4} sin (θ) + sin^{3} (θ)

- \frac{1}{4} (3 sin (θ) - 4 sin^{3} (θ))

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = - \frac{1}{4}, b = 3 sin (θ), c = 4 sin^{3} (θ)

= - \frac{1}{4} \cdot 3 sin (θ) - (- \frac{1}{4}) \cdot 4 sin^{3} (θ)

Aplicar as regras dos sinais

- (- a) = a

= - 3 \cdot \frac{1}{4} sin (θ) + 4 \cdot \frac{1}{4} sin^{3} (θ)

Simplificar

- 3 \cdot \frac{1}{4} sin (θ) + 4 \cdot \frac{1}{4} sin^{3} (θ) : - \frac{3}{4} sin (θ) + sin^{3} (θ)

- 3 \cdot \frac{1}{4} sin (θ) + 4 \cdot \frac{1}{4} sin^{3} (θ)

3 \cdot \frac{1}{4} sin (θ) = \frac{3}{4} sin (θ)

3 \cdot \frac{1}{4} sin (θ)

Multiplicar frações:

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

= \frac{1 \cdot 3}{4} sin (θ)

Multiplicar os números:

1 \cdot 3 = 3

= \frac{3}{4} sin (θ)

4 \cdot \frac{1}{4} sin^{3} (θ) = sin^{3} (θ)

4 \cdot \frac{1}{4} sin^{3} (θ)

Multiplicar frações:

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

= \frac{1 \cdot 4}{4} sin^{3} (θ)

Eliminar o fator comum:

4

= sin^{3} (θ) \cdot 1

Multiplicar:

sin^{3} (θ) \cdot 1 = sin^{3} (θ)

= sin^{3} (θ)

= - \frac{3}{4} sin (θ) + sin^{3} (θ)

= - \frac{3}{4} sin (θ) + sin^{3} (θ)

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

Somar elementos similares:

\frac{3}{4} sin (θ) - \frac{3}{4} sin (θ) = 0

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

Fatorar o termo comum

sin (θ)

= sin (θ) (\frac{3}{4} - \frac{3}{4})

\frac{3}{4} - \frac{3}{4} = 0

\frac{3}{4} - \frac{3}{4}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{3 - 3}{4}

Simplificar

= 0

= 0

= sin^{3} (θ)

= sin^{3} (θ)

= sin^{3} (θ)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

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