已知x∈(0,π/2),求证:sinx+cosx>1
问题描述:
已知x∈(0,π/2),求证:sinx+cosx>1
答
sinx+cosx=2^0.5sin(x+π/4)
x∈(0,π/2)时,x+π/4∈(π/4,3π/4),(2^0.5)/2
答
sinx+cosx
=√2(√2/2*sinx+√2/2*cosx)
=√2(sinxcosπ/4+cosxsinπ/4)
=√2sin(x+π/4)
π/4
sin(x+π/4)最小=√2/2
这里都不取到
所以sin(x+π/4)>√2/2
所以√2sin(x+π/4)>1
所以sinx+cosx>1