设a=(0,π/3),β∈(π/6,π/2),且α,β满足5倍根号3sina+5cosa=8,根号2sinβ+根号6cosβ=2求cos(a+π/6)的值求cos(a+β)的值
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![设a=(0,π/3),β∈(π/6,π/2),且α,β满足5倍根号3sina+5cosa=8,根号2sinβ+根号6cosβ=2求cos(a+π/6)的值求cos(a+β)的值](/uploads/image/z/7198526-38-6.jpg?t=%E8%AE%BEa%3D%280%2C%CF%80%2F3%29%2C%CE%B2%E2%88%88%28%CF%80%2F6%2C%CF%80%2F2%29%2C%E4%B8%94%CE%B1%2C%CE%B2%E6%BB%A1%E8%B6%B35%E5%80%8D%E6%A0%B9%E5%8F%B73sina%2B5cosa%3D8%2C%E6%A0%B9%E5%8F%B72sin%CE%B2%2B%E6%A0%B9%E5%8F%B76cos%CE%B2%3D2%E6%B1%82cos%EF%BC%88a%2B%CF%80%2F6%EF%BC%89%E7%9A%84%E5%80%BC%E6%B1%82cos%EF%BC%88a%2B%CE%B2%EF%BC%89%E7%9A%84%E5%80%BC)
设a=(0,π/3),β∈(π/6,π/2),且α,β满足5倍根号3sina+5cosa=8,根号2sinβ+根号6cosβ=2求cos(a+π/6)的值求cos(a+β)的值
设a=(0,π/3),β∈(π/6,π/2),且α,β满足5倍根号3sina+5cosa=8,根号2sinβ+根号6cosβ=2
求cos(a+π/6)的值
求cos(a+β)的值
设a=(0,π/3),β∈(π/6,π/2),且α,β满足5倍根号3sina+5cosa=8,根号2sinβ+根号6cosβ=2求cos(a+π/6)的值求cos(a+β)的值
5倍根号3sina+5cosa=8
(根号3/2)sina+(1/2)cosa=4/5
sin(π/3)sina+cos(π/3)cosa=4/5
cos(a-π/3)=4/5
因为a∈(0,π/3),所以a-π/3∈(-π/3,0)
所以sin(a-π/3)=-根号(1-(cos(a-π/3))^2)=-根号(1-(4/5)^)=-3/5
根号2sinβ+根号6cosβ=2
2根号2((1/2)sinb+(根号3/2)cosb)=2
(1/2)sinb+(根号3/2)cosb=根号2/2
sin(π/6)sinb+cos(π/6)cosb=根号2/2
cos(b-π/6)=根号2/2
因为β∈(π/6,π/2)
所以b-π/6∈(0,π/3)
所以sin(b-π/6)=根号(1-(cos(b-π/6))^2)=根号(1-(根号2/2)^2)=根号2/2
(1)cos(a+π/6)=cos(a-π/3+π/2)=cos(a-π/3)cos(π/2)-sin(a-π/3)sin(π/2)
=(4/5)*0-(-3/5)*1=3/5
因为a∈(0,π/3),所以a+π/6∈(π/6,π/2)
所以sin(a+π/6)=根号(1-(cos(a+π/6))^2)=根号(1-(3/5)^2)=4/5
(2)cos(a+b)=cos(a+π/6+b-π/6)=cos(a+π/6)cos(b-π/6)-sin(a+π/6)sin(b-π/6)
=3/5*(根号2/2)-4/5*(根号2/2)=- 根号2/10