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Popular Trigonometria >

provar cos(x+pi/3)+cos(x-pi/3)=cos(x)

Solução

provar

cos (x + \frac{π}{3}) + cos (x - \frac{π}{3}) = cos (x)

Solução

Verdadeiro

Passos da solução

cos (x + \frac{π}{3}) + cos (x - \frac{π}{3}) = cos (x)

Manipular o lado direito

cos (x + \frac{π}{3}) + cos (x - \frac{π}{3})

Reeecreva usando identidades trigonométricas

cos (x - \frac{π}{3})

Use a identidade de diferença de ângulos:

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

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

Simplificar

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

cos (x) cos (\frac{π}{3}) + sin (x) sin (\frac{π}{3})

Simplificar

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{3}) = \frac{1}{2}

cos (x)

tabela de periodicidade com ciclo de

2 πn

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

Simplificar

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{3}) = \frac{3}{2}

sin (x)

tabela de periodicidade com ciclo de

2 πn

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}

= \frac{3}{2}

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

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

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

Reeecreva usando identidades trigonométricas

cos (x + \frac{π}{3})

Use a identidade de soma de ângulos:

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

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

Simplificar

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

cos (x) cos (\frac{π}{3}) - sin (x) sin (\frac{π}{3})

Simplificar

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{3}) = \frac{1}{2}

cos (x)

tabela de periodicidade com ciclo de

2 πn

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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} 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} cos (x) - sin (\frac{π}{3}) sin (x)

Simplificar

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{3}) = \frac{3}{2}

sin (x)

tabela de periodicidade com ciclo de

2 πn

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}

= \frac{3}{2}

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

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

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

Simplificar

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

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

Agrupar termos semelhantes

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

Somar elementos similares:

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

\frac{1}{2} cos (x) + \frac{1}{2} cos (x)

Fatorar o termo comum

cos (x)

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

\frac{1}{2} + \frac{1}{2} = 1

\frac{1}{2} + \frac{1}{2}

Aplicar a regra

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

= \frac{1 + 1}{2}

Simplificar

= 1

= cos (x)

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

Somar elementos similares:

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

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

Fatorar o termo comum

sin (x)

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

- \frac{3}{2} + \frac{3}{2} = 0

- \frac{3}{2} + \frac{3}{2}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{- 3 + 3}{2}

Fatorar

- 3 + 3 : 0

- 3 + 3

Fatorar o termo comum

3

= 3 (- 1 + 1)

Simplificar

= 0

= \frac{0}{2}

Aplicar a regra

\frac{0}{a} = 0, a \neq = 0

= 0

= 0

= cos (x)

= cos (x)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

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