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

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. \(2\)