30. (Udesc-SC) Assinale a alternativa que corresponde ao valor da expressão:
$$
6 \cos ^{2}\left(\frac{13 \pi}{6}\right)-4 \cos ^{2}\left(\frac{11 \pi}{4}\right)+\operatorname{sen}\left(-\frac{7 \pi}{6}\right)+\operatorname{tg}^{2}\left(\frac{31 \pi}{3}\right)
$$
a) 6
b) 5
c) $ \frac{9}{2} $
d) 3
e) $ \frac{23}{4} $

Question

Accepted Answer

Para resolver a expressão dada, vamos calcular cada termo separadamente:

$$
6 \cos ^{2}\left(\frac{13 \pi}{6}\right)-4 \cos ^{2}\left(\frac{11 \pi}{4}\right)+\operatorname{sen}\left(-\frac{7 \pi}{6}\right)+\operatorname{tg}^{2}\left(\frac{31 \pi}{3}\right)
$$

1. Calculando $ \cos \left(\frac{13 \pi}{6}\right) $:

$$
\frac{13 \pi}{6} = 2 \pi + \frac{\pi}{6}
$$

Como $ \cos(2 \pi + x) = \cos(x) $, temos:

$$
\cos \left(\frac{13 \pi}{6}\right) = \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
$$

Portanto:

$$
\cos^2 \left(\frac{13 \pi}{6}\right) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}
$$

Logo:

$$
6 \cos^2 \left(\frac{13 \pi}{6}\right) = 6 \cdot \frac{3}{4} = \frac{18}{4} = \frac{9}{2}
$$

2. Calculando $ \cos \left(\frac{11 \pi}{4}\right) $:

$$
\frac{11 \pi}{4} = 2 \pi + \frac{3 \pi}{4}
$$

Como $ \cos(2 \pi + x) = \cos(x) $, temos:

$$
\cos \left(\frac{11 \pi}{4}\right) = \cos \left(\frac{3 \pi}{4}\right) = -\frac{\sqrt{2}}{2}
$$

Portanto:

$$
\cos^2 \left(\frac{11 \pi}{4}\right) = \left(-\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2}
$$

Logo:

$$
-4 \cos^2 \left(\frac{11 \pi}{4}\right) = -4 \cdot \frac{1}{2} = -2
$$

3. Calculando $ \operatorname{sen} \left(-\frac{7 \pi}{6}\right) $:

$$
-\frac{7 \pi}{6} = -\pi - \frac{\pi}{6}
$$

Como $ \operatorname{sen}(-\pi - x) = -\operatorname{sen}(x) $, temos:

$$
\operatorname{sen} \left(-\frac{7 \pi}{6}\right) = -\operatorname{sen} \left(\frac{\pi}{6}\right) = -\frac{1}{2}
$$

4. Calculando $ \operatorname{tg} \left(\frac{31 \pi}{3}\right) $:

$$
\frac{31 \pi}{3} = 10 \pi + \frac{\pi}{3}
$$

Como $ \operatorname{tg}(10 \pi + x) = \operatorname{tg}(x) $, temos:

$$
\operatorname{tg} \left(\frac{31 \pi}{3}\right) = \operatorname{tg} \left(\frac{\pi}{3}\right) = \sqrt{3}
$$

Portanto:

$$
\operatorname{tg}^2 \left(\frac{31 \pi}{3}\right) = (\sqrt{3})^2 = 3
$$

Agora somamos todos os termos calculados:

$$
\frac{9}{2} - 2 - \frac{1}{2} + 3
$$

Simplificando:

$$
\frac{9}{2} - \frac{4}{2} - \frac{1}{2} + 3 = \frac{9}{2} - \frac{5}{2} + 3 = \frac{4}{2} + 3 = 2 + 3 = 5
$$

**Resposta: B**

A expressão avaliada resulta em 5.

30. (Udesc-SC) Assinale a alternativa que corresponde ao val...

Ask a question and receive an answer instantly