Calcule 2 arcsen (−√2/2) + 12 arccos (√3/2). Mostre os cálculos com as devidas justificativas.

Vamos calcular passo a passo a expressão: \[ 2 \arcsin\left(-\frac{\sqrt{2}}{2}\right) + 12 \arccos\left(\frac{\sqrt{3}}{2}\right) \] --- 1. Calcular \(\arcsin\left(-\frac{\sqrt{2}}{2}\right)\): - Sabemos que \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). - Como o valor é negativo, \(\arcsin\left(-\frac{\sqrt{2}}{2}\right) = -\frac{\pi}{4}\), pois o arco seno varia de \(-\frac{\pi}{2}\) a \(\frac{\pi}{2}\). --- 2. Calcular \(\arccos\left(\frac{\sqrt{3}}{2}\right)\): - Sabemos que \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\). - O arco cosseno varia de 0 a \(\pi\), então \(\arccos\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}\). --- 3. Substituir e calcular: \[ 2 \times \left(-\frac{\pi}{4}\right) + 12 \times \frac{\pi}{6} = -\frac{2\pi}{4} + \frac{12\pi}{6} = -\frac{\pi}{2} + 2\pi = \frac{3\pi}{2} \] --- Resposta final: \[ 2 \arcsin\left(-\frac{\sqrt{2}}{2}\right) + 12 \arccos\left(\frac{\sqrt{3}}{2}\right) = \frac{3\pi}{2} \]