作业帮1.已知γ=2β=4α≠0,且α∈[0,2п],且sin²β=sinαsinγ≠0,求α、β、γ.2.已知α、β均为锐角,且3sin2α-2sin2β=0,3sin²α+2sin ²β=1,求证：α+2β=90
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![1.已知γ=2β=4α≠0,且α∈[0,2п],且sin²β=sinαsinγ≠0,求α、β、γ.2.已知α、β均为锐角,且3sin2α-2sin2β=0,3sin²α+2sin ²β=1,求证：α+2β=90](/uploads/image/z/1159309-37-9.jpg?t=1.%E5%B7%B2%E7%9F%A5%CE%B3%3D2%CE%B2%3D4%CE%B1%E2%89%A00%2C%E4%B8%94%CE%B1%E2%88%88%5B0%2C2%D0%BF%5D%2C%E4%B8%94sin%26sup2%3B%CE%B2%3Dsin%CE%B1sin%CE%B3%E2%89%A00%2C%E6%B1%82%CE%B1%E3%80%81%CE%B2%E3%80%81%CE%B3.2.%E5%B7%B2%E7%9F%A5%CE%B1%E3%80%81%CE%B2%E5%9D%87%E4%B8%BA%E9%94%90%E8%A7%92%2C%E4%B8%943sin2%CE%B1-2sin2%CE%B2%3D0%2C3sin%26sup2%3B%CE%B1%2B2sin+%26sup2%3B%CE%B2%3D1%2C%E6%B1%82%E8%AF%81%EF%BC%9A%CE%B1%2B2%CE%B2%3D90)
1.已知γ=2β=4α≠0,且α∈[0,2п],且sin²β=sinαsinγ≠0,求α、β、γ.2.已知α、β均为锐角,且3sin2α-2sin2β=0,3sin²α+2sin ²β=1,求证：α+2β=90
1.已知γ=2β=4α≠0,且α∈[0,2п],且sin²β=sinαsinγ≠0,
求α、β、γ.
2.已知α、β均为锐角,且3sin2α-2sin2β=0,3sin²α+2sin ²β=1,
求证：α+2β=90
1.已知γ=2β=4α≠0,且α∈[0,2п],且sin²β=sinαsinγ≠0,求α、β、γ.2.已知α、β均为锐角,且3sin2α-2sin2β=0,3sin²α+2sin ²β=1,求证：α+2β=90
(1)
r=2b=4a
sinb^2=sin2a^2=4sina^2cosa^2=4(1-cosa^2)cosa^2
sinαsinγ
=sinasin4a
=sina(2sin2acos2a)
=sina[4sinacosa(2cosa^2-1)]
=4sina^2cosa(2cosa^2-1)
=4(1-cosa^2)cosa(2cosa^2-1)
sin²β=sinαsinγ
4(1-cosa^2)cosa^2=4(1-cosa^2)cosa(2cosa^2-1)
4(1-cosa^2)cosa(cosa-1)(2cosa+1)=0
cosa=1或-1或0或-1/2
由于sin²β=sinαsinγ≠0
所以cosa=-1/2
由于α∈[0,2п]
a=2/3*Pi,b=2a=4/3*Pi,r=4a=8/3*Pi
或者
a=4/3*Pi,b=2a=8/3*Pi,r=4a=16/3*Pi
(2)
3sin²α+2sin ²β=1
3sina^2+2*(1-cos2b)/2=1
3sina^2=cos2b (1)
3sin2α-2sin2β=0
3/2*sin2a=3sinacosa=sin2b (2)
(1)^2+(2)^2得：
9sina^2cosa^2+9sina^4=1
9sina^2(sina^2+cosa^2)=1
sina^2=1/9
由于α、β均为锐角
则sina=1/3
由(1)得：
cos2b=3sina^2=3*(1/9)=1/3
所以cos2b=sina=1/3
所以
α+2β=90
证毕