Question

数学题，求解，第16题

Answer 1

16、f(x)=cos²x+sinxcosx
=(cos2x+1)/2+1/2sin2x
=1/2cos2x+1/2sin2x+1/2
=1/√2(√2/2cos2x+√2/2sin2x)+1/2
=√2/2sin(2x+π/4)+1/2
（1）f(π/6)=√2/2sin(2*π/6+π/4)+1/2
=√2/2sin(π/3+π/4)+1/2
=√2/2sin(7π/12)+1/2
（2）α∈(π/2,π)
sinα=3/5
cosα=-√(1-sin²α)
=-√(1-(3/5)²)
=-4/5
f(α/2+π/24)=√2/2sin(2*(α/2+π/24)+π/4)+1/2
=√2/2sin(α+π/12+π/4)+1/2
=√2/2sin(α+π/3)+1/2
=0.7071*0.9659+0.5
=1.1830
=√2/2(sinαcosπ/3+cosαsinπ/3)+1/2
=√2/2(3/5*1/2+(-4/5)*√3/2)+1/2
=√2/2(3/10-2√3/5)+1/2
=3√2/20-√6/5+1/2