化简sin1度6次方+ sin2度6次方 ····sin89度6次方=
1个回答
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]
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