高二物理-三角函數

..........................
1.設15度<=x<=75度，求Sin^4 x+Cos^4 x的最大，最小值
Sol
Sin^4 x+Cos^4 x
=(Sin^2 x+Cos^2 x)^2-2Sin^2 xCos^2 x
=1－(1/2)*(2SinxCosx)^2
=1－(1/2)Sin^2 (2x)
15度<=x<=75度
30度<=2x<=150度
Sin30度=1/2，Sin9０度=1,Sin150度=1/2
1/2<=Sin(2x)<=1
1/4<=Sin^2 (2x)<=1
1/8<=(1/2)Sin^2 (2x)<=1/2
－1/2<=－(1/2)Sin^2(2x)<=－1/8
1/2<=1－(1/2)Sin^2(2x)<=7/8
1/2<=Sin^4 x+Cos^4 x<=7/8

2.設@=180度，化簡Sin^4 @/8+Sin^4 3@/8+Sin^45@/8+Sin^4 7@/8之值為?
Sol
Sin^4 @/8+Sin^4 3@/8+Sin^4 5@/8+Sin^4 7@/8
=Sin^4 @/8+Sin^4 7@/8+Sin^4 3@/8+Sin^4 5@/8
=Sin^4 @/8+Sin^4 @/8+Sin^4 3@/8+Sin^4 3@/8
=2Sin^4 @/8+2Sin^4 3@/8
Sin^2 22.5度=(1－Cos45度)/2=(2－Ccos45度)/4=(2－√2)/4
Sin^4 22.5度=(6－4√2)/4
Sin^2 67.5度=(1－Cos135度)/2=(2－2Cos135度)/4=(2+√2)/4
Sin^4 67.5度=(6+4√2)/16
Sin^4 @/8+Sin^4 3@/8+Sin^4 5@/8+Sin^4 7@/8
=2*(12/16)
=3/2