Question

化简sin1度6次方+ sin2度6次方 ····sin89度6次方=

Answer 1

先求(sinα)^6+(cosα)^6 的值

(sinα)^6+(cosα)^6

=[(sinα)^2+(cosα)^2][(sinα)^4-(sinα)^2(cosα)^2+(cosα)^4]

=1[(sin²α+cos²α)²-3sin²αcos²α]

=1-3/4(4sinαcosα)²

=1-3/4(sin²2α)

=1-3/8(1-cos4α)

=5/8+3/8*cos4α

∴（sin1º)^6+(sin89º)^6

=（sin1º)^6+(cos1º)^6

=5/8+3/8 cos4º

即（sin1º)^6+(sin89º)^6=5/8+3/8 cos4º

同理：（sin2º)^6+(sin88º)^6=5/8+3/8 cos8º

（sin3º)^6+(sin87º)^6=5/8+3/8 cos12º

.

（sin44º)^6+(sin46º)^6=5/8+3/8 cos176º

（sin45º)^6=(√2/2)^6=1/8

将上面45个等式相加:

即

sin1度6次方+ sin2度6次方 ····sin89度6次方

=5/8*44+1/8+3/8( cos4º+cos8º+cos12º+.+cos168º+cos172º+cos176º)

=221/8

[ cos4º+cos8º+cos12º+.+cos168º+cos172º+cos176º=0]