Calcule o valor exato
Trigonometria: Infinito 11 A - Parte 1 Pág. 96 Ex. 54
Enunciado
Calcule o valor exato de cada uma das expressões.
- $sen\,\frac{13\pi }{3}+\cos 5\pi -tg\,(-7\pi )+\cos (-\frac{23\pi }{4})$
- $se{{n}^{2}}\,(-\frac{7\pi }{4})+{{\cos }^{2}}(-\frac{7\pi }{4})$
- $sen\,\frac{19\pi }{3}+\cos (-3\pi )-tg\,(-\frac{15\pi }{4})+\cos (-\frac{11\pi }{6})$
- $tg\,\frac{13\pi }{4}+\cos 6\pi -sen\,(-\frac{7\pi }{2})+\cos (-\frac{17\pi }{3})$
Resolução
- Ora,
\[\begin{array}{*{35}{l}}
sen\,\frac{13\pi }{3}+\cos 5\pi -tg\,(-7\pi )+\cos (-\frac{23\pi }{4}) & = & sen\,(4\pi +\frac{\pi }{3})+\cos (4\pi +\pi )-tg\,(-7\pi +0)+\cos (-6\pi +\frac{\pi }{4}) \\
{} & = & sen\,\frac{\pi }{3}+\cos \pi -tg\,0+\cos \frac{\pi }{4} \\
{} & = & \frac{\sqrt{3}}{2}-1-0+\frac{\sqrt{2}}{2} \\
{} & = & \frac{\sqrt{3}+\sqrt{2}-2}{2} \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
se{{n}^{2}}\,(-\frac{7\pi }{4})+{{\cos }^{2}}(-\frac{7\pi }{4}) & = & se{{n}^{2}}\,(-2\pi +\frac{\pi }{4})+{{\cos }^{2}}(-2\pi +\frac{\pi }{4}) \\
{} & = & se{{n}^{2}}\,(\frac{\pi }{4})+{{\cos }^{2}}(\frac{\pi }{4}) \\
{} & = & {{\left( \frac{\sqrt{2}}{2} \right)}^{2}}+{{\left( \frac{\sqrt{2}}{2} \right)}^{2}} \\
{} & = & 1 \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
sen\,\frac{19\pi }{3}+\cos (-3\pi )-tg\,(-\frac{15\pi }{4})+\cos (-\frac{11\pi }{6}) & = & sen\,(6\pi +\frac{\pi }{3})+\cos (-4\pi +\pi )-tg\,(-4\pi +\frac{\pi }{4})+\cos (-2\pi +\frac{\pi }{6}) \\
{} & = & \frac{\sqrt{3}}{2}-1-1+\frac{\sqrt{3}}{2} \\
{} & = & \sqrt{3}-2 \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
tg\,\frac{13\pi }{4}+\cos 6\pi -sen\,(-\frac{7\pi }{2})+\cos (-\frac{17\pi }{3}) & = & tg\,\frac{\pi }{4}+1-sen\,(-4\pi +\frac{\pi }{2})+\cos (-6\pi +\frac{\pi }{3}) \\
{} & = & 1+1-1+\frac{1}{2} \\
{} & = & \frac{3}{2} \\
\end{array}\]
