设a>0,0<b<π/2,且a+b=5π/6,求函数y=2-sin^a - cos^b的值域
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设a>0,0<b<π/2,且a+b=5π/6,求函数y=2-sin^a - cos^b的值域
![设a>0,0<b<π/2,且a+b=5π/6,求函数y=2-sin^a - cos^b的值域](/uploads/image/z/8070536-56-6.jpg?t=%E8%AE%BEa%EF%BC%9E0%2C0%EF%BC%9Cb%EF%BC%9C%CF%80%2F2%2C%E4%B8%94a%2Bb%3D5%CF%80%2F6%2C%E6%B1%82%E5%87%BD%E6%95%B0y%3D2-sin%5Ea+-+cos%5Eb%E7%9A%84%E5%80%BC%E5%9F%9F)
y=2-sin²a-cos²b
2y=4-2sin²a-2cos²b
=4-[1-cos2a]-[1+cos2b]
=2+cos2a-cos2b
=2-2sin(a+b)sin(a-b)
=2-2sin(5π/6)sin(150º-2b)
=2-sin(150-2b)
∴2-2y=sin(150º-2b).
∵0<b<90º
∴-30º<150-2b<150º
∴-1/2<sin(150º-2b)≤1
即-1/2<2-2y≤1
∴1/2≤y<5/4
∴值域为[1/2, 5/4)
2y=4-2sin²a-2cos²b
=4-[1-cos2a]-[1+cos2b]
=2+cos2a-cos2b
=2-2sin(a+b)sin(a-b)
=2-2sin(5π/6)sin(150º-2b)
=2-sin(150-2b)
∴2-2y=sin(150º-2b).
∵0<b<90º
∴-30º<150-2b<150º
∴-1/2<sin(150º-2b)≤1
即-1/2<2-2y≤1
∴1/2≤y<5/4
∴值域为[1/2, 5/4)
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