Question

5．求值：sin1°sin3°sin5°…sin87°sin89°．

Answer 1

分析 利用誘導(dǎo)公式與倍角公式即可得出．

解答 解：設(shè)T=sin1°sin3°sin5°…sin87°sin89°．
則T=（sin1°sin89°）（sin3°sin87°）•…•（sin43°sin47°）sin45°
=$\frac{1}{2}$sin2°•$\frac{1}{2}sin{6}^{°}$•…•$\frac{1}{2}$sin86°$•\frac{\sqrt{2}}{2}$
=$\frac{1}{{2}^{22}}$×$\frac{\sqrt{2}}{2}$×cos4°cos8°…cos84°cos88°
=$\frac{1}{{2}^{22}}$×$\frac{\sqrt{2}}{2}$×$\frac{2sin{4}^{°}cos{4}^{°}}{2sin{4}^{°}}$•cos8°…cos84°cos88°
=$\frac{1}{{2}^{22}}$×$\frac{\sqrt{2}}{2}$×$\frac{1}{2sin{4}^{°}}sin{8}^{°}$•cos8°…cos84°cos88°
=$\frac{1}{{2}^{22}}$×$\frac{\sqrt{2}}{2}$×$\frac{sin17{6}^{°}}{{2}^{22}sin{4}^{°}}$
=$\frac{\sqrt{2}}{{2}^{45}}$．

點(diǎn)評(píng) 本題考查了誘導(dǎo)公式與倍角公式，考查了推理能力與計(jì)算能力，屬于中檔題．