Calcule o valor de \( \cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) \). Resposta: \( \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} \).

Para calcular \( \cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) \), podemos usar a fórmula do cosseno da soma: \[ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \] Aqui, \( a = \frac{\pi}{3} \) e \( b = \frac{\pi}{4} \). 1. Cálculo dos valores de \( \cos \) e \( \sin \): - \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \) - \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) - \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) - \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) 2. Substituindo na fórmula: \[ \cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \left(\frac{1}{2} \cdot \frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}\right) \] 3. Simplificando: \[ = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} \] Portanto, a resposta correta é: \[ \cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} \] A sua resposta está correta!