设a＞0,0＜b＜π/2,且a+b=5π/6,求函数y=2-sin^a - cos^b的值域
问题描述：
设a＞0,0＜b＜π/2,且a+b=5π/6,求函数y=2-sin^a - cos^b的值域
答
y=2-sina-cos(5π/6 - a) = 2-sina +√3/2 cosa-1/2 sina = 2 - 3/2sina + √3/2 cosa
函数y的值域为：{ y | 2-√3 ≤ y ≤ 2+√3 }
答
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)