高一数学设向量a=(cos(α+β),sin(α+β))设向量a=(cos(α+β),sin(α+β)),b=(cos(α-β),sin(α-β),且a+b=(4/5,3/5).(1)求tanα(2)求2cos^2(α/2)-3sinα-1/根号2sin(α+π/4).
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![高一数学设向量a=(cos(α+β),sin(α+β))设向量a=(cos(α+β),sin(α+β)),b=(cos(α-β),sin(α-β),且a+b=(4/5,3/5).(1)求tanα(2)求2cos^2(α/2)-3sinα-1/根号2sin(α+π/4).](/uploads/image/z/6847819-43-9.jpg?t=%E9%AB%98%E4%B8%80%E6%95%B0%E5%AD%A6%E8%AE%BE%E5%90%91%E9%87%8Fa%3D%28cos%28%CE%B1%2B%CE%B2%29%2Csin%28%CE%B1%2B%CE%B2%29%29%E8%AE%BE%E5%90%91%E9%87%8Fa%3D%28cos%28%CE%B1%2B%CE%B2%29%2Csin%28%CE%B1%2B%CE%B2%29%29%2Cb%3D%28cos%28%CE%B1-%CE%B2%EF%BC%89%2Csin%28%CE%B1-%CE%B2%EF%BC%89%2C%E4%B8%94a%2Bb%3D%284%2F5%2C3%2F5%29.%EF%BC%881%EF%BC%89%E6%B1%82tan%CE%B1%EF%BC%882%EF%BC%89%E6%B1%822cos%5E2%28%CE%B1%2F2%EF%BC%89-3sin%CE%B1-1%2F%E6%A0%B9%E5%8F%B72sin%28%CE%B1%2B%CF%80%2F4%29.)
高一数学设向量a=(cos(α+β),sin(α+β))设向量a=(cos(α+β),sin(α+β)),b=(cos(α-β),sin(α-β),且a+b=(4/5,3/5).(1)求tanα(2)求2cos^2(α/2)-3sinα-1/根号2sin(α+π/4).
高一数学设向量a=(cos(α+β),sin(α+β))
设向量a=(cos(α+β),sin(α+β)),b=(cos(α-β),sin(α-β),且a+b=(4/5,3/5).
(1)求tanα
(2)求2cos^2(α/2)-3sinα-1/根号2sin(α+π/4).
高一数学设向量a=(cos(α+β),sin(α+β))设向量a=(cos(α+β),sin(α+β)),b=(cos(α-β),sin(α-β),且a+b=(4/5,3/5).(1)求tanα(2)求2cos^2(α/2)-3sinα-1/根号2sin(α+π/4).
(1)a=(cos(α+β),sin(α+β)),b=(cos(α-β),sin(α-β),且a+b=(4/5,3/5).
而a+b=(cos(α+β)+,cos(α-β),sin(α+β)+sin(α-β))
=(2cosαcosβ,2sinαcosβ)
所以2cosαcosβ=4/5,2sinαcosβ=3/5
两式相除得:tanα=3/4
(2)[2cos^2(α/2)-3sinα-1]/[√2sin(α+π/4)]
=(cosα-3sinα)/(sinα+cosα)
=(1-3tanα)/(tanα+1)
=(1-9/4)/(3/4+1)=-5/7
|2a+b|^2=|a-2b|^2 4a^2 +4ab+b^2=a^2-4ab+4b^2 3a^2+8ab-7b^2=0 设α-β=-θ 3a^2+8|a||b|cosθ-7b^2=0 ∵a^2=1 b^
由cos(α+β)+cos(α-β)=4/5 得cosαcosβ=2/5①
sin(α+β)+sin(α-β)=3/5 得sinαcosβ=3/10②
②/①=tanα=3/4
(2)原式=﹙cosα-3sinα﹚/﹙sinα+cosα﹚=1-3tanα/1+tanα=-5/7