Question

Cos (arcsin 7/25 - arccos 12/13)

questão de trigonometria ajuda aí pfv tá valendo 1 ponto ​

Answer 1

Resposta: 323/325.

Explicação passo a passo:

Calcular cos(arcsin 7/25 − arccos 12/13).

Sejam α = arcsin 7/25 e β = arccos 12/13. Queremos calcular cos(α − β). Aplicamos a fórmula da expansão do cosseno de uma diferença:

    $\cos(\alpha-\beta)=\cos \alpha\cos\beta+\sin \alpha \sin \beta\\\\\\=\cos(\arcsin \frac{7}{25})\cdot \cos(\arccos\frac{12}{13})+\sin(\arcsin\frac{7}{25})\sin(\arccos\frac{12}{13})\\\\\\=\cos(\arcsin \frac{7}{25})\cdot \dfrac{12}{13}+\dfrac{7}{25}\cdot \sin(\arccos\frac{12}{13})\qquad (i)$

Como α = arcsin(7/25), então devemos ter sen α = 7/25. Para calcular cos α, podemos utilizar a relação trigonométrica fundamental:

    $\cos^2\alpha+\sin^2\alpha=1\\\\\\\Longrightarrow\quad \cos^2\alpha=1-\sin^2\alpha\\\\\\\Longrightarrow\quad \cos^2\alpha=1-\left(\dfrac{7}{25}\right)^{\!2}\\\\\\\Longrightarrow\quad \cos^2\alpha=1-\dfrac{49}{625}\\\\\\\Longrightarrow\quad \cos^2\alpha=\dfrac{625-49}{625}\\\\\\\Longrightarrow\quad \cos^2\alpha=\dfrac{576}{625}$

    $\Longrightarrow\quad \cos\alpha=\sqrt{\dfrac{576}{625}}\\\\\\\Longrightarrow\quad \cos\alpha=\dfrac{24}{25}\\\\\\\Longrightarrow\quad \cos(\arcsin\frac{7}{25})=\dfrac{24}{25}\qquad (ii)$

De modo análogo, se β = arccos 12/13, então cos β = 12/13. Logo,

    $\sin^2\beta=1-\cos^2\beta\\\\\Longrightarrow\quad \sin^2\beta=1-\left(\dfrac{12}{13}\right)^{\!2}\\\\\\\Longrightarrow\quad \sin^2\beta=1-\dfrac{144}{169}\\\\\\\Longrightarrow\quad \sin^2\beta=\dfrac{169-144}{169}$

    $\Longrightarrow\quad \sin^2\beta=\dfrac{25}{169}\\\\\\\Longrightarrow\quad \sin\beta=\sqrt{\dfrac{25}{169}}\\\\\\\Longrightarrow\quad \sin\beta=\dfrac{5}{13}}\\\\\\ \Longrightarrow\quad \sin(\arccos \frac{12}{13})=\dfrac{5}{13}}\qquad (iii)$

Substituindo (ii) e (iii) em (i), chegamos ao resultado:

    $=\dfrac{24}{25}\cdot \dfrac{12}{13}+\dfrac{7}{25}\cdot \dfrac{5}{13}\\\\\\=\dfrac{288}{325}+\dfrac{35}{325}\\\\\\=\dfrac{323}{325}\quad\longleftarrow\quad \mathsf{resposta.}$

Dúvidas? Comente.

Bons estudos!