Question

求tan3/5π的函数值

Answer 1

解：令 x=cos(π/5),
y=cos(2π/5).
则 y=2x^2 -1,
即 2x^2 -y -1=0. (1)
因为 cos(4π/5)=cos(2 *2π/5)
=2y^2-1,
且 cos(4π/5)=cos(π-π/5)
= -x,
所以 -x=2y^2-1.
即 2y^2 +x -1=0. (2)
(1)-(2)得
2(x^2-y^2)-(y+x)=0,
即 (x+y)(2x-2y-1)=0.
又因为 x>0, y>0,
所以 2x-2y-1=0,
即 y=x -1/2. (3).
把(3)代入(1), 得
2x^2 -(x -1/2) -1=0.
解得
x1=(1 +根号5)/4,
x2=(1 -根号5)/4. (舍去).
把 x=(1 +根号5)/4 代入(3)得
y=(根号5 -1)/4.
即 cos(2π/5)=(根号5 -1)/4.
所以 sec(2π/5)=1 /cos(2π/5)
= 根号5 +1.
又因为 [sec(2π/5)]^2 -[tan(2π/5)]^2=1,
所以 tan(2π/5)= 根号[ (根号5 +1)^2 -1]
= 根号[ 5+2(根号5) ].
所以 tan(3π/5)= tan(π-2π/5)
= -根号[ 5+2(根号5) ].

= = = = = = = = =
计算可能有误。你最好检查一下。
如果设 x=tan (π/5) ,y=tan (2π/5), 要解一个二元三次方程.
用 [sec(2π/5)]^2 -[tan(2π/5)]^2=1,是因为有二重根号.
先算sin (2π/5) 会有二重根号出现.
再除于cos (2π/5) ,根号更麻烦.
15度, 75度, 18度, 36度, 72度都是特殊角（可用根号表示）.

Answer 2

tan(π*3÷5)=0.03291055548
tan(3÷5π）=0.003.333345679

Answer 3

0.03291

Answer 4

tan(3pi/5)=-3.07768