Question

Calcule as expressões:
a) tg 5pi/4 + cos 5pi/6
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sen 7pi/6


b)sen3pi/2 + 2 cos pi
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3sen pi/2

Answer 1

Para facilitar o cálculo das seguintes expressões, é recomendável transformar para um arco notável do 1º quadrante.

$\text{a) \,\,} \frac{\text{tg}\left(\frac{5\pi}{4} \right)+\cos\left(\frac{5\pi}{6} \right )}{\text{sen}\left(\frac{7\pi}{6} \right )}\\ \\ =\frac{\text{tg}\left(\frac{5\pi}{4}-\pi \right )-\cos\left(\pi-\frac{5\pi}{6} \right )}{-\text{sen}\left(\frac{7\pi}{6}-\pi \right )}\\ \\ =\frac{\text{tg}\left(\frac{5\pi-4\pi}{4} \right )-\cos\left(\frac{6\pi-5\pi}{6} \right )}{-\text{sen}\left(\frac{7\pi-6\pi}{6} \right )}\\ \\ =\frac{\text{tg}\left(\frac{\pi}{4} \right )-\cos\left(\frac{\pi}{6} \right )}{-\text{sen}\left(\frac{\pi}{6} \right )}\\ \\ =\frac{1-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} \rightarrow \left( \frac{\times 2}{\times 2} \right )\\ \\ =\frac{2-\sqrt{3}}{-1}\\ \\ =\boxed{\sqrt{3}-2}$


$\text{b) \,\,} \frac{\text{sen}\left(\frac{3\pi}{2} \right )+2\cos\left(\pi \right )}{3 \text{sen}\left(\frac{\pi}{2} \right )}\\ \\ =\frac{-\text{sen}\left(\frac{3\pi}{2}-\pi \right )-2\cos\left(\pi-\pi \right )}{3 \cdot 1}\\ \\ =\frac{-\text{sen}\left(\frac{3\pi-2\pi}{2})-2\cos\left(0 \right )}{3}\\ \\ =\frac{-\text{sen}\left(\frac{\pi}{2} \right )-2 \cdot 1}{3}\\ \\ =\frac{-1-2}{3}\\ \\ =\frac{-3}{3}\\ \\ =\boxed{-1 }$