已知5根号3sinx+5cosx=6,根号2siny+根号6cosy=1且x属于(0,π/3),y属于(π/6,π/2)
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已知5根号3sinx+5cosx=6,根号2siny+根号6cosy=1且x属于(0,π/3),y属于(π/6,π/2),求cos(x+y)
![已知5根号3sinx+5cosx=6,根号2siny+根号6cosy=1且x属于(0,π/3),y属于(π/6,π/2)](/uploads/image/z/18730207-55-7.jpg?t=%E5%B7%B2%E7%9F%A55%E6%A0%B9%E5%8F%B73sinx%2B5cosx%3D6%2C%E6%A0%B9%E5%8F%B72siny%2B%E6%A0%B9%E5%8F%B76cosy%3D1%E4%B8%94x%E5%B1%9E%E4%BA%8E%280%2C%CF%80%2F3%29%2Cy%E5%B1%9E%E4%BA%8E%28%CF%80%2F6%2C%CF%80%2F2%29)
5√3sinx+5cosx=6,
5√3sinx/10+5cosx/10=6/10
√3/2*sinx+1/2*cosx=3/5
sinxcosπ/6+cosxsinπ/6=3/5
sin(x+π/6)=3/5
x∈(0,π/3)
x+π/6∈(π/6,π/2)
cos(x+π/6)=4/5
√2siny+√6cosy=1
√2siny/√2+√6cosy/√2=1/√2
siny+√3cosy=√2/2
1/2*siny+√3/2*cosy=√2/4
sinysinπ/6+cosycosπ/6=√2/4
cos(y-π/6)=√2/4
y∈(π/6,π/2)
y-π/6∈(0,π/3)
sin(y-π/6)=√14/4
cos(x+y)
=cos[(x+π/6)+(y-π/6)]
=cos(x+π/6)cos(y-π/6)-sin(x+π/6)sin(y-π/6)
=4/5*√2/4-3/5*√14/4
=√2/5-3√14/12
5√3sinx/10+5cosx/10=6/10
√3/2*sinx+1/2*cosx=3/5
sinxcosπ/6+cosxsinπ/6=3/5
sin(x+π/6)=3/5
x∈(0,π/3)
x+π/6∈(π/6,π/2)
cos(x+π/6)=4/5
√2siny+√6cosy=1
√2siny/√2+√6cosy/√2=1/√2
siny+√3cosy=√2/2
1/2*siny+√3/2*cosy=√2/4
sinysinπ/6+cosycosπ/6=√2/4
cos(y-π/6)=√2/4
y∈(π/6,π/2)
y-π/6∈(0,π/3)
sin(y-π/6)=√14/4
cos(x+y)
=cos[(x+π/6)+(y-π/6)]
=cos(x+π/6)cos(y-π/6)-sin(x+π/6)sin(y-π/6)
=4/5*√2/4-3/5*√14/4
=√2/5-3√14/12
x,y∈[0,π/6],(cosx)*(2^x)+2^(π/2)=a*(2^x),且(根号3)*cosy+siny=2a
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