Question

Trigonometria




1-Calcule o valor da expressão P= $cos \frac{35\pi }{4} + sen \frac{-43\pi }{6}$


2-Imagem em anexo

Answer 1

1)

$\displaystyle \text{cos}(\frac{35\pi}{4})+\text{sen}(\frac{-43\pi}{6})$

Vamos lembrar que o seno é uma função ímpar, ou seja, sen(-x) = -sen(x).

Daí :

$\displaystyle \text{cos}(\frac{35\pi}{4})-\text{sen}(\frac{43\pi}{6}) \\\\\\ \text{cos}(\frac{32\pi}{4}+\frac{3\pi}{4}) - \text{sen}(\frac{36\pi}{6}+\frac{7\pi}{6}) \\\\\\ \text{cos}(8\pi+\frac{3\pi}{4})-\text{sen}(6\pi+ \frac{7\pi}{6}) \\\\\\\ \underline{\text{Temos que}}: \\\\ \text{cos}(8\pi+\frac{3\pi}{4})=\text{cos}(\frac{3\pi}{4})=-\text{cos}(\frac{\pi}{4}) \to \text{(Redu{\c c}{\~a}o ao primeiro quadrante)} \\\\ \text{sen}(6\pi+\frac{7\pi}{6}) = \text{sen}(\frac{7\pi}{6})=-\text{sen}(\frac{\pi}{6})$

Daí :

$\displaystyle -\text{cos}(\frac{\pi}{4}) -\text{sen}(\frac{\pi}{6}) =\frac{-\sqrt 2}{2}-\frac{1}{2} \\\\\\ \underline{\text{Portanto}}: \\\\\\ \text{cos}(\frac{35\pi}{4})+\text{sen}(\frac{-43\pi}{6}) = \huge\boxed{\frac{-\sqrt{2}-1}{2}\ }\checkmark$

2)

$\displaystyle \text{cos x = ? } \ ; \ \text{sen x}=\frac{4}{5} \ \ , \ \frac{\pi}{2}<\text x<\pi \\\\\\ \text{Rela{\c c}{\~a}o fundamental da trigonometria} : \\\\ \text{sen}^2\text x+\text{cos}^2 \text x=1\\\\ (\frac{4}{5})^2+\text{cos}^2\text x=1 \\\\ \text{cos}^2\text x= 1-\frac{16}{25} \\\\ \text{cos x}=\pm \sqrt{\frac{25-16}{25}} \\\\\\ \text{cos x}=\pm\sqrt{\frac{9}{25}} \\\\\\ \huge\boxed{\text{cos x}=\frac{-3}{5}\ }\checkmark$

Negativo porque x está no 2 º quadrante e o cosseno lá é negativo.