Question

Considere a função S(x)=1+2senx+4(senx)^2+8(senx)^3,para x € R .Determine o valor de S(pi/4).

Answer 1

$S(x) = 1 + 2 \cdot (\sin x) + 4 \cdot (\sin x)^2 + 8 \cdot (\sin x)^3 \\\\\\S\left(\frac{\pi}{4}\right)=1+2\cdot\left[\sin\left(\frac{\pi}{4}\right)\right]+4\cdot\left[\sin\left(\frac{\pi}{4}\right)\right]^2+8\cdot\left[\sin\left(\frac{\pi}{4}\right)\right]^3$

 Sabemos que $\sin \left ( \frac{\pi}{4} \right ) = \frac{\sqrt{2}}{2}$. Daí,

$S \left ( \frac{\pi}{4} \right ) = 1 + 2 \cdot \frac{\sqrt{2}}{2} + 4 \cdot \left ( \frac{\sqrt{2}}{2} \right )^2 + 8 \cdot \left ( \frac{\sqrt{2}}{2} \right )^3 \\\\\\ S \left ( \frac{\pi}{4} \right ) = 1 + 2 \cdot \frac{\sqrt{2}}{2} + 4 \cdot \frac{2}{4} + 8 \cdot \frac{2\sqrt{2}}{8} \\\\\\ S \left ( \frac{\pi}{4} \right ) = 1 + \sqrt{2} + 2 + 2\sqrt{2} \\\\\\ \boxed{S \left ( \frac{\pi}{4} \right ) = 3(1 + \sqrt{2})}$