### + / -檢視主題

#ed_op#DIV#ed_cl#這題是我好久以前問的, 題目快忘了......#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#原來要用三倍角公式#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#(怪不得得問sin#ed_op#SUP#ed_cl#-1#ed_op#/SUP#ed_cl#x)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl#
galaxylee 寫到:#ed_op#DIV#ed_cl##ed_op#DIV#ed_cl#16x^4-8x^3-12x^2+8x-1=0#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#(2x-1)(8x^3-6x+1)=0#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#x=1/2是一根#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#8x^3-6x+1=0 =&gt; 2(3x-4x^3)=1#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#令x=sinθ&nbsp;上式變成 2sin3θ=1 =&gt; sin3θ=1/2#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#3θ=(π/6)+2kπ，或(5π/6)+2kπ，其中k為整數#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#θ=(π/18)+(2kπ)/3，或(5π/18)+(2kπ)/3#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#可歸納出sinθ=sin(π/18) 或 sin(5π/18)，sin(-7π/18)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#所以方程式的四根為 x = 1/2(=sin(π/6))、sin(π/18)、sin(5π/18)、sin(-7π/18)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#故 sin^(-1)x = π/6、π/18、5π/18、-7π/18#ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl#
#ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#是否需先證(8x^3-6x+1)=0之三根皆在-1~1#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#,才可令 x = s in@#ed_op#/DIV#ed_cl#
#ed_op#DIV#ed_cl##ed_op#DIV#ed_cl#16x^4-8x^3-12x^2+8x-1=0#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#(2x-1)(8x^3-6x+1)=0#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#x=1/2是一根#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#8x^3-6x+1=0 =&gt; 2(3x-4x^3)=1#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#令x=sinθ&nbsp;上式變成 2sin3θ=1 =&gt; sin3θ=1/2#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#3θ=(π/6)+2kπ，或(5π/6)+2kπ，其中k為整數#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#θ=(π/18)+(2kπ)/3，或(5π/18)+(2kπ)/3#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#可歸納出sinθ=sin(π/18) 或 sin(5π/18)，sin(-7π/18)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#所以方程式的四根為 x = 1/2(=sin(π/6))、sin(π/18)、sin(5π/18)、sin(-7π/18)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#故 sin^(-1)x = π/6、π/18、5π/18、-7π/18#ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl#

### [問題]三角方程

#ed_op#DIV#ed_cl#設一方程: 16x#ed_op#SUP#ed_cl#4#ed_op#/SUP#ed_cl#-8x#ed_op#SUP#ed_cl#3#ed_op#/SUP#ed_cl#-12x#ed_op#SUP#ed_cl#2#ed_op#/SUP#ed_cl#+8x-1=0#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#試求sin#ed_op#SUP#ed_cl#-1#ed_op#/SUP#ed_cl#x =??#ed_op#/DIV#ed_cl#