Question

Calcule o valor da expressão a seguir:

Answer 1

O resultado dessa expressão é:

          $\Large\displaystyle\text{ \begin{gathered} \sf\frac{\sin\left(\frac{3\pi}{2}\right)+\cos(\pi )+5\sin \left(\frac{\pi}{2}\right)}{3\cos\left(\frac{\pi}{2} \right)+\cos (2\pi )-\sin (2\pi )} = 3\end{gathered} }$

Desejamos calcular a seguinte expressão:

$\Large\displaystyle\text{ \begin{gathered} \sf\frac{\sin\left(\frac{3\pi}{2}\right)+\cos(\pi )+5\sin \left(\frac{\pi}{2}\right)}{3\cos\left(\frac{\pi}{2} \right)+\cos (2\pi )-\sin (2\pi )} \end{gathered} }$

Passando de radianos para graus, temos que;

$\Large\displaystyle\text{ \begin{gathered} \sf\frac{\sin\left(270^{\circ}\right)+\cos(180^{\circ} )+5\sin \left(90^{\circ}\right)}{3\cos\left(90^{\circ}\right)+\cos (360^{\circ} )-\sin (360^{\circ} )} \end{gathered} }$

Agora, com ajuda do ciclo trigonométrico ( em anexo ), temos que:

$\Large\displaystyle\text{ \begin{gathered} \sf\frac{\sin\left(\frac{3\pi}{2}\right)+\cos(\pi )+5\sin \left(\frac{\pi}{2}\right)}{3\cos\left(\frac{\pi}{2} \right)+\cos (2\pi )-\sin (2\pi )} = \frac{-1-1+5(1)}{3(0)+1-0} \end{gathered} }$

$\Large\displaystyle\text{ \begin{gathered} \sf\frac{\sin\left(\frac{3\pi}{2}\right)+\cos(\pi )+5\sin \left(\frac{\pi}{2}\right)}{3\cos\left(\frac{\pi}{2} \right)+\cos (2\pi )-\sin (2\pi )} = \frac{-2+5}{1} \end{gathered} }$

$\Large\displaystyle\text{ \begin{gathered} \therefore \green{\underline{\boxed{ \sf\frac{\sin\left(\frac{3\pi}{2}\right)+\cos(\pi )+5\sin \left(\frac{\pi}{2}\right)}{3\cos\left(\frac{\pi}{2} \right)+\cos (2\pi )-\sin (2\pi )} = 3}}}\ \ \ (\checkmark).\end{gathered} }$

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