Question

4．若tanα=2tan$\frac{π}{5}$，则$\frac{cos（α-\frac{3π}{10}）}{sin（α-\frac{π}{5}）}$=3．

Answer 1

分析 利用三角恒等变换化简要求的式子，可得结果．

解答 解：∵tanα=2tan$\frac{π}{5}$，则$\frac{cos（α-\frac{3π}{10}）}{sin（α-\frac{π}{5}）}$=$\frac{cosαcos\frac{3π}{10}+sinαsin\frac{3π}{10}}{sinαcos\frac{π}{5}-cosαsin\frac{π}{5}}$=$\frac{cos\frac{3π}{10}+tanα•sin\frac{3π}{10}}{tanα•cos\frac{π}{5}-sin\frac{π}{5}}$=$\frac{cos\frac{3π}{10}+2\frac{sin\frac{π}{5}}{cos\frac{π}{5}}•sin\frac{3π}{10}}{2tan\frac{π}{5}•cos\frac{π}{5}-sin\frac{π}{5}}$
=$\frac{cos\frac{3π}{10}+2tan\frac{π}{5}sin\frac{3π}{10}}{2sin\frac{π}{5}-sin\frac{π}{5}}$=$\frac{cos\frac{π}{5}cos\frac{3π}{10}+2sin\frac{π}{5}•sin\frac{3π}{10}}{sin\frac{π}{5}•cos\frac{π}{5}}$=$\frac{cos（\frac{3π}{10}-\frac{π}{5}）+sin\frac{π}{5}•sin\frac{3π}{10}}{sin\frac{π}{5}•cos\frac{π}{5}}$ 
=$\frac{cos\frac{π}{10}-\frac{1}{2}[cos（\frac{3π}{10}-\frac{π}{5}）-cos（\frac{3π}{10}+\frac{π}{5}）]}{\frac{1}{2}sin\frac{2π}{5}}$=$\frac{\frac{3}{2}cos\frac{π}{10}}{\frac{1}{2}sin（\frac{π}{2}-\frac{π}{10}）}$=3，
故答案为：3．

点评 本题主要考查三角恒等变换及化简求值，属于中档题．