已知向量m=(cosa,sina)和n=(根号2-sina,cosa),a∈(π,2π),且|m+n|=(8根号2)/5,求cos(a/2+π/8)的值
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![已知向量m=(cosa,sina)和n=(根号2-sina,cosa),a∈(π,2π),且|m+n|=(8根号2)/5,求cos(a/2+π/8)的值](/uploads/image/z/572260-4-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fm%3D%EF%BC%88cosa%2Csina%29%E5%92%8Cn%3D%28%E6%A0%B9%E5%8F%B72-sina%2Ccosa%29%2Ca%E2%88%88%EF%BC%88%CF%80%2C2%CF%80%29%2C%E4%B8%94%7Cm%2Bn%7C%3D%EF%BC%888%E6%A0%B9%E5%8F%B72%EF%BC%89%2F5%2C%E6%B1%82cos%EF%BC%88a%2F2%2B%CF%80%2F8%29%E7%9A%84%E5%80%BC)
已知向量m=(cosa,sina)和n=(根号2-sina,cosa),a∈(π,2π),且|m+n|=(8根号2)/5,求cos(a/2+π/8)的值
已知向量m=(cosa,sina)和n=(根号2-sina,cosa),a∈(π,2π),且|m+n|=(8根号2)/5,求cos(a/2+π/8)的值
已知向量m=(cosa,sina)和n=(根号2-sina,cosa),a∈(π,2π),且|m+n|=(8根号2)/5,求cos(a/2+π/8)的值
m+n=(cosa-sina+√2,sina+cosa)
∴|m+n|²
=(cosa-sina+√2)²+(sina+cosa)²
=(cosa-sina)²+2√2(cosa-sina)+2+(cosa+sina)²
=2(sin²a+cos²a)+2-2√2(sina-cosa)
=2+2-2√2(sina-cosa)
=4-2√2(sina-cosa)
=2+2-4[(√2/2)sina-(√2/2)cosa]
=2+2-4[sinπ/4sina-cosπ/4cosa]
=4+4[cosπ/4cosa-sinπ/4sina]
=4+4cos(a+π/4)
4+4cos(a+π/4)=(8√2/5)²
4+4cos(a+π/4)=128/25
4cos(a+π/4)=28/25
cos(a+π/4)=7/25
cos(a+π/4)=2cos²(a/2+π/8)-1
2cos²(a/2+π/8)-1=7/25
cos²(a/2+π/8)=16/25
cos(a/2+π/8)=±4/5
a∈(π,2π)
a/2∈(π/2,π)
a/2+π/8∈(5π/8,9π/8)
∴cos(a/2+π/8)=-4/5
m+n=(cosa-sina+√2,sina+cosa)
∴|m+n|²
=(cosa-sina+√2)²+(sina+cosa)²
=(cosa-sina)²+2√2(cosa-sina)+2+(cosa+sina)²
=2(sin²a+cos²a)+2-2√2(sina-cosa)
=...
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m+n=(cosa-sina+√2,sina+cosa)
∴|m+n|²
=(cosa-sina+√2)²+(sina+cosa)²
=(cosa-sina)²+2√2(cosa-sina)+2+(cosa+sina)²
=2(sin²a+cos²a)+2-2√2(sina-cosa)
=2+2-2√2(sina-cosa)
=4-2√2(sina-cosa)
=2+2-4[(√2/2)sina-(√2/2)cosa]
=2+2-4[sinπ/4sina-cosπ/4cosa]
=4+4[cosπ/4cosa-sinπ/4sina]
=4+4cos(a+π/4)
4+4cos(a+π/4)=(8√2/5)²
4+4cos(a+π/4)=128/25
4cos(a+π/4)=28/25
cos(a+π/4)=7/25
cos(a+π/4)=2cos²(a/2+π/8)-1
2cos²(a/2+π/8)-1=7/25
cos²(a/2+π/8)=16/25
cos(a/2+π/8)=±4/5
a∈(π,2π)
a/2∈(π/2,π)
a/2+π/8∈(5π/8,9π/8)
∴cos(a/2+π/8)=-4/5
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