d) $$ \frac{3}{5} \operatorname{cotg} \frac{5 \pi}{3}-\frac{6}{7} \operatorname{cotg} \frac{7 \pi}{6}-\frac{2}{3} \operatorname{sen} \frac{3 \pi}{2}+\frac{4}{5} \cos \frac{5 \pi}{4} $$

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UpStudy AI · Accepted Answer

Vamos calcular a expressão passo a passo:

$$ \frac{3}{5} \cot \frac{5\pi}{3} - \frac{6}{7} \cot \frac{7\pi}{6} - \frac{2}{3} \sin \frac{3\pi}{2} + \frac{4}{5} \cos \frac{5\pi}{4} $$

### 1. Cálculo de $$ \cot \frac{5\pi}{3} $$:

$$
\cot \theta = \frac{\cos \theta}{\sin \theta}
$$

$$
\frac{5\pi}{3} \text{ corresponde a } 300^\circ
$$

$$
\cos \frac{5\pi}{3} = \frac{1}{2}
$$
$$
\sin \frac{5\pi}{3} = -\frac{\sqrt{3}}{2}
$$
$$
\cot \frac{5\pi}{3} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
$$
$$
\frac{3}{5} \cot \frac{5\pi}{3} = \frac{3}{5} \times \left(-\frac{\sqrt{3}}{3}\right) = -\frac{\sqrt{3}}{5}
$$

### 2. Cálculo de $$ \cot \frac{7\pi}{6} $$:

$$
\frac{7\pi}{6} \text{ corresponde a } 210^\circ
$$

$$
\cos \frac{7\pi}{6} = -\frac{\sqrt{3}}{2}
$$
$$
\sin \frac{7\pi}{6} = -\frac{1}{2}
$$
$$
\cot \frac{7\pi}{6} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}
$$
$$
-\frac{6}{7} \cot \frac{7\pi}{6} = -\frac{6}{7} \times \sqrt{3} = -\frac{6\sqrt{3}}{7}
$$

### 3. Cálculo de $$ \sin \frac{3\pi}{2} $$:

$$
\frac{3\pi}{2} \text{ corresponde a } 270^\circ
$$

$$
\sin \frac{3\pi}{2} = -1
$$
$$
-\frac{2}{3} \sin \frac{3\pi}{2} = -\frac{2}{3} \times (-1) = \frac{2}{3}
$$

### 4. Cálculo de $$ \cos \frac{5\pi}{4} $$:

$$
\frac{5\pi}{4} \text{ corresponde a } 225^\circ
$$

$$
\cos \frac{5\pi}{4} = -\frac{\sqrt{2}}{2}
$$
$$
\frac{4}{5} \cos \frac{5\pi}{4} = \frac{4}{5} \times \left(-\frac{\sqrt{2}}{2}\right) = -\frac{2\sqrt{2}}{5}
$$

### 5. Somando todos os termos:

$$
-\frac{\sqrt{3}}{5} - \frac{6\sqrt{3}}{7} + \frac{2}{3} - \frac{2\sqrt{2}}{5}
$$

### 6. Aproximação Numérica:

Para obter uma aproximação numérica, substituímos os valores de $$ \sqrt{2} $$ e $$ \sqrt{3} $$:

$$
\sqrt{2} \approx 1,4142 \quad \text{e} \quad \sqrt{3} \approx 1,7320
$$

$$
-\frac{1,7320}{5} \approx -0,3464
$$
$$
-\frac{6 \times 1,7320}{7} \approx -1,4846
$$
$$
\frac{2}{3} \approx 0,6667
$$
$$
-\frac{2 \times 1,4142}{5} \approx -0,5657
$$

Somando os termos aproximados:

$$
-0,3464 - 1,4846 + 0,6667 - 0,5657 \approx -1,73
$$

### **Resultado Final:**

A expressão avaliada é aproximadamente $$-1,73$$.
A expressão $$ \frac{3}{5} \cot \frac{5\pi}{3} - \frac{6}{7} \cot \frac{7\pi}{6} - \frac{2}{3} \sin \frac{3\pi}{2} + \frac{4}{5} \cos \frac{5\pi}{4} $$ é aproximadamente igual a $$-1,73$$.

d)

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