已知向量a=(1-tanx,1),b=(1+sin2x+cos2x,0)已知a=(1-tanx,1),b=(1+sin2x+cos2x,0).记函数f(X)=a*b,(1)求函数f(X)的解析式.(2)f(α+π/8)=√2/5,且α属于(0,π/2).求f(α)(a,b都是向量)
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/27 18:19:22
![已知向量a=(1-tanx,1),b=(1+sin2x+cos2x,0)已知a=(1-tanx,1),b=(1+sin2x+cos2x,0).记函数f(X)=a*b,(1)求函数f(X)的解析式.(2)f(α+π/8)=√2/5,且α属于(0,π/2).求f(α)(a,b都是向量)](/uploads/image/z/1265447-47-7.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%281-tanx%2C1%29%2Cb%3D%281%EF%BC%8Bsin2x%2Bcos2x%2C0%29%E5%B7%B2%E7%9F%A5a%3D%281-tanx%2C1%29%2Cb%3D%281%2Bsin2x%2Bcos2x%2C0%29.%E8%AE%B0%E5%87%BD%E6%95%B0f%28X%29%3Da%2Ab%2C%EF%BC%881%EF%BC%89%E6%B1%82%E5%87%BD%E6%95%B0f%28X%29%E7%9A%84%E8%A7%A3%E6%9E%90%E5%BC%8F.%EF%BC%882%EF%BC%89f%28%CE%B1%2B%CF%80%2F8%EF%BC%89%3D%E2%88%9A2%2F5%2C%E4%B8%94%CE%B1%E5%B1%9E%E4%BA%8E%EF%BC%880%2C%CF%80%2F2%EF%BC%89.%E6%B1%82f%28%CE%B1%EF%BC%89%EF%BC%88a%2Cb%E9%83%BD%E6%98%AF%E5%90%91%E9%87%8F%EF%BC%89)
已知向量a=(1-tanx,1),b=(1+sin2x+cos2x,0)已知a=(1-tanx,1),b=(1+sin2x+cos2x,0).记函数f(X)=a*b,(1)求函数f(X)的解析式.(2)f(α+π/8)=√2/5,且α属于(0,π/2).求f(α)(a,b都是向量)
已知向量a=(1-tanx,1),b=(1+sin2x+cos2x,0)
已知a=(1-tanx,1),b=(1+sin2x+cos2x,0).记函数f(X)=a*b,(1)求函数f(X)的解析式.(2)f(α+π/8)=√2/5,且α属于(0,π/2).求f(α)(a,b都是向量)
已知向量a=(1-tanx,1),b=(1+sin2x+cos2x,0)已知a=(1-tanx,1),b=(1+sin2x+cos2x,0).记函数f(X)=a*b,(1)求函数f(X)的解析式.(2)f(α+π/8)=√2/5,且α属于(0,π/2).求f(α)(a,b都是向量)
(1)
因为sin2x=2sinxcosx,cos2x=2cos²x-1
所以f(x)=ab=(1-tanx)(1+sin2x+cos2x)=(1-tanx)(1+2sinxcosx+2cos²-1)
=(1-tanx)(2cos²x-2sinxcosx)=(sinx+cosx)(2cos²x-2sinxcosx)/cosx
=2(sinx+cosx)(cosx-sinx)=2(cos²x-sin²x)=2cos(2x)
(2)
f(α+π/8)=√2/5即2cos(2α+π/4)=√2/5
cos2α=4/5,sin2α=3/5或cos2α=-3/5,sin2α=-4/5
因为α属于(0,π/2),所以2α属于(0,π),所以sin2α>0,
故sin2α=3/5,cos2α=4/5
所以f(α)=2cos(2α)=8/5
111