y=sin^x+根号3sinx*cosx的最小正周期和单调区间

题目

y=sin^x+根号3sinx*cosx的最小正周期和单调区间

答案

y = sin²x + √3sinxcosx
= (1 - cos2x)/2 + (√3/2)sin2x
= (√3/2)sin2x - (1/2)cos2x + 1/2
= sin(2x - π/6) + 1/2
蕞小正周期：2π/2 = π
2kπ - π/2 < 2x - π/6 < 2kπ + π/2
kπ - π/6 < x < kπ + π/3
单增区间：(kπ - π/6,kπ + π/3),k∈Z
2kπ + π/2 < 2x - π/6 < 2kπ + 3π/2
kπ + π/3 < x < kπ + 5π/6
单减区间：(kπ + π/3,kπ + 5π/6),k∈Z