以知f (x) = -1/2 + (sin 5x/2)/(2*sin x/2) ,x∈ (0,∏),写出f (x)关于
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以知f (x) = -1/2 + (sin 5x/2)/(2*sin x/2) ,x∈ (0,∏),写出f (x)关于 cos x的表达式,并求f (x)的最小
![以知f (x) = -1/2 + (sin 5x/2)/(2*sin x/2) ,x∈ (0,∏),写出f (x)关于](/uploads/image/z/19518071-23-1.jpg?t=%E4%BB%A5%E7%9F%A5f+%28x%29+%3D+-1%2F2+%2B+%28sin+5x%2F2%29%2F%282%2Asin+x%2F2%29+%2Cx%E2%88%88+%280%2C%E2%88%8F%29%2C%E5%86%99%E5%87%BAf+%28x%29%E5%85%B3%E4%BA%8E)
f(x)=-1/2+sin5/2x/2sinx/2,x∈(0,∏)
=-1/2+sinx/2cos2x+cosx/2sin2x/2sinx/2
=-1/2+sinx/2(1-cos^2x)+2sinx/2*(1-cosx)/2
1-2cos^2x+1-cosx-1/2
=-cos^2-cosx/2+1/2
=9/10-(cosx+1/4)^2
∵x∈(0,∏)
∴cosx∈(-1,1)
∴当cosx=1时
f(x)min=9/10-25/16=-1
=-1/2+sinx/2cos2x+cosx/2sin2x/2sinx/2
=-1/2+sinx/2(1-cos^2x)+2sinx/2*(1-cosx)/2
1-2cos^2x+1-cosx-1/2
=-cos^2-cosx/2+1/2
=9/10-(cosx+1/4)^2
∵x∈(0,∏)
∴cosx∈(-1,1)
∴当cosx=1时
f(x)min=9/10-25/16=-1
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