$$ \text { Exercícios de aplicação } $$ 1. Calcule o valor numérico das expressões que se seguem, utilizando a tabela dos valores trigonométricos: $$ \frac{6}{2} \operatorname{sen}\left(\frac{3 \pi}{2}\right) \times \cos \left(360^{\circ}\right)-\operatorname{cotg} g^{2}\left(30^{\circ}\right) $$ $$ \frac{24}{4 \times 3} \cos ^{2}\left(45^{\circ}\right)+\frac{\operatorname{cotg}\left(60^{\circ}\right)}{\operatorname{tg}\left(30^{\circ}\right)} $$ $$ \frac{\operatorname{sen}\left(270^{\circ}\right) \times \operatorname{cotg}^{2}\left(270^{\circ}\right)+\cos \left(360^{\circ}\right) \times \operatorname{cotg}\left(45^{\circ}\right)}{\frac{4}{3} \operatorname{sen} 2\left(30^{\circ}\right) \times \operatorname{tg}^{2}\left(45^{\circ}\right)-\operatorname{tg}\left(30^{\circ}\right) \times \frac{1}{\cos \left(30^{\circ}\right)}} $$ $$ \frac{16^{\operatorname{sen}\left(30^{\circ}\right)}+\operatorname{cotg}^{\cot \left(90^{\circ}\right)}}{2 \cos ^{2}\left(30^{\circ}\right)+3 \operatorname{cotg}^{2}\left(60^{\circ}\right)} $$

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Claro! Vamos resolver cada uma das expressões passo a passo utilizando os valores trigonométricos conhecidos.

### 1. $$\frac{6}{2} \cdot \sen\left(\frac{3\pi}{2}\right) \times \cos\left(360^{\circ}\right) - \cotg^{2}\left(30^{\circ}\right)$$

**Passo 1:** Simplificar os coeficientes:
$$
\frac{6}{2} = 3
$$

**Passo 2:** Calcular $$\sen\left(\frac{3\pi}{2}\right)$$:
$$
\frac{3\pi}{2} = 270^{\circ} \Rightarrow \sen(270^{\circ}) = -1
$$

**Passo 3:** Calcular $$\cos(360^{\circ})$$:
$$
\cos(360^{\circ}) = 1
$$

**Passo 4:** Calcular $$\cotg\left(30^{\circ}\right)$$:
$$
\tan(30^{\circ}) = \frac{\sqrt{3}}{3} \Rightarrow \cotg(30^{\circ}) = \frac{1}{\tan(30^{\circ})} = \sqrt{3}
$$
$$
\cotg^{2}\left(30^{\circ}\right) = (\sqrt{3})^{2} = 3
$$

**Passo 5:** Substituir na expressão original:
$$
3 \times (-1) \times 1 - 3 = -3 - 3 = -6
$$

**Resultado:** $$-6$$

---

### 2. $$\frac{24}{4 \times 3} \cdot \cos^{2}\left(45^{\circ}\right) + \frac{\cotg\left(60^{\circ}\right)}{\tg\left(30^{\circ}\right)}$$

**Passo 1:** Simplificar os coeficientes:
$$
\frac{24}{4 \times 3} = \frac{24}{12} = 2
$$

**Passo 2:** Calcular $$\cos^{2}\left(45^{\circ}\right)$$:
$$
\cos(45^{\circ}) = \frac{\sqrt{2}}{2} \Rightarrow \cos^{2}(45^{\circ}) = \left(\frac{\sqrt{2}}{2}\right)^{2} = \frac{2}{4} = 0,5
$$
$$
2 \times 0,5 = 1
$$

**Passo 3:** Calcular $$\cotg\left(60^{\circ}\right)$$ e $$\tg\left(30^{\circ}\right)$$:
$$
\tan(60^{\circ}) = \sqrt{3} \Rightarrow \cotg(60^{\circ}) = \frac{1}{\tan(60^{\circ})} = \frac{1}{\sqrt{3}} \approx 0,577
$$
$$
\tan(30^{\circ}) = \frac{\sqrt{3}}{3} \approx 0,577
$$
$$
\frac{\cotg\left(60^{\circ}\right)}{\tg\left(30^{\circ}\right)} = \frac{0,577}{0,577} = 1
$$

**Passo 4:** Somar os resultados:
$$
1 + 1 = 2
$$

**Resultado:** $$2$$

---

### 3. $$\frac{\sen\left(270^{\circ}\right) \times \cotg^{2}\left(270^{\circ}\right) + \cos\left(360^{\circ}\right) \times \cotg\left(45^{\circ}\right)}{\frac{4}{3} \sen\left(60^{\circ}\right) \times \tg^{2}\left(45^{\circ}\right) - \tg\left(30^{\circ}\right) \times \frac{1}{\cos\left(30^{\circ}\right)}}$$

**Numerador:**

**Passo 1:** Calcular $$\sen\left(270^{\circ}\right)$$:
$$
\sen(270^{\circ}) = -1
$$

**Passo 2:** Calcular $$\cotg\left(270^{\circ}\right)$$:
$$
\cotg(270^{\circ}) = \frac{\cos(270^{\circ})}{\sen(270^{\circ})} = \frac{0}{-1} = 0
$$
$$
\cotg^{2}\left(270^{\circ}\right) = 0^{2} = 0
$$

**Passo 3:** Calcular $$\cos\left(360^{\circ}\right)$$ e $$\cotg\left(45^{\circ}\right)$$:
$$
\cos(360^{\circ}) = 1
$$
$$
\cotg(45^{\circ}) = 1
$$

**Passo 4:** Substituir no numerador:
$$
(-1) \times 0 + 1 \times 1 = 0 + 1 = 1
$$

**Denominador:**

**Passo 1:** Calcular $$\sen\left(60^{\circ}\right)$$ e $$\tg^{2}\left(45^{\circ}\right)$$:
$$
\sen(60^{\circ}) = \frac{\sqrt{3}}{2} \approx 0,866
$$
$$
\tg(45^{\circ}) = 1 \Rightarrow \tg^{2}(45^{\circ}) = 1
$$
$$
\frac{4}{3} \times 0,866 \times 1 \approx 1,155
$$

**Passo 2:** Calcular $$\tg\left(30^{\circ}\right)$$ e $$\cos\left(30^{\circ}\right)$$:
$$
\tg(30^{\circ}) = \frac{\sqrt{3}}{3} \approx 0,577
$$
$$
\cos(30^{\circ}) = \frac{\sqrt{3}}{2} \approx 0,866
$$
$$
\frac{1}{\cos(30^{\circ})} = \frac{2}{\sqrt{3}} \approx 1,1547
$$
$$
\tg\left(30^{\circ}\right) \times \frac{1}{\cos\left(30^{\circ}\right)} \approx 0,577 \times 1,1547 \approx 0,666
$$

**Passo 3:** Subtrair os termos:
$$
1,155 - 0,666 \approx 0,489
$$

**Passo 4:** Dividir numerador pelo denominador:
$$
\frac{1}{0,489} \approx 2,048
$$

**Resultado:** Aproximadamente $$2,048$$

---

### 4. $$\frac{16^{\sen\left(30^{\circ}\right)} + \cotg^{\cot\left(90^{\circ}\right)}}{2 \cos^{2}\left(30^{\circ}\right) + 3 \cotg^{2}\left(60^{\circ}\right)}$$

**Numerador:**

**Passo 1:** Calcular $$16^{\sen\left(30^{\circ}\right)}$$:
$$
\sen(30^{\circ}) = 0,5 \Rightarrow 16^{0,5} = \sqrt{16} = 4
$$

**Passo 2:** Calcular $$\cotg^{\cot\left(90^{\circ}\right)}$$:
$$
\tg(90^{\circ}) = \infty \Rightarrow \cotg(90^{\circ}) = 0
$$
$$
\cotg^{0} = 1
$$

**Passo 3:** Somar os termos:
$$
4 + 1 = 5
$$

**Denominador:**

**Passo 1:** Calcular $$\cos^{2}\left(30^{\circ}\right)$$:
$$
\cos(30^{\circ}) = \frac{\sqrt{3}}{2} \Rightarrow \cos^{2}(30^{\circ}) = \left(\frac{\sqrt{3}}{2}\right)^{2} = \frac{3}{4} = 0,75
$$
$$
2 \times 0,75 = 1,5
$$

**Passo 2:** Calcular $$\cotg^{2}\left(60^{\circ}\right)$$:
$$
\tg(60^{\circ}) = \sqrt{3} \Rightarrow \cotg(60^{\circ}) = \frac{1}{\sqrt{3}} \approx 0,577
$$
$$
\cotg^{2}(60^{\circ}) = (0,577)^{2} \approx 0,333
$$
$$
3 \times 0,333 \approx 1
$$

**Passo 3:** Somar os termos:
$$
1,5 + 1 = 2,5
$$

**Passo 4:** Dividir numerador pelo denominador:
$$
\frac{5}{2,5} = 2
$$

**Resultado:** $$2$$

---

**Resumo dos Resultados:**
1. $$-6$$
2. $$2$$
3. Aproximadamente $$2,048$$
4. $$2$$
1. $$-6$$ 2. $$2$$ 3. Aproximadamente $$2,048$$ 4. $$2$$

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