解不等式arccos(cosx)>arcsin(sinx)
问题描述：
解不等式arccos(cosx)>arcsin(sinx)
请详解
答
令x=2kπ+t,t∈[0,2π)
当x=2kπ,2kπ+π/2时,arccos(cosx)=arcsin(sinx)
当x=2kπ+π,2kπ+3π/2时,arccos(cosx)>arcsin(sinx)
(1)、当t∈(0,π/2)时,
arccos(cosx)=t∈(0,π/2)
arcsin(sinx)=t∈(0,π/2)
arccos(cosx)=arcsin(sinx)
(2)、当t∈(π/2,π)时,
arccos(cosx)=t∈(π/2,π)
arcsin(sinx)=π-t∈(0,π/2)
arccos(cosx)>arcsin(sinx)
(3)、当t∈(π,3π/2)时
arccos(cosx)=2π-t∈(π/2,π)
arcsin(sinx)=π-t∈(-π/2,0)
arccos(cosx)>arcsin(sinx)
(4)、当t∈(3π/2,2π)时,
arccos(cosx)=2π-t∈(0,π/2)
arcsin(sinx)=t-2π∈(-π/2,0)
arccos(cosx)>arcsin(sinx)
综上所诉,
x∈(2kπ+π/2,2kπ+2π)