Question

sinα+cosα的最值

Answer 1

因为 (sin α +cos α)^2 = (sin α)^2 +(cos α)^2 +2 sin α cos α

= 1 +sin 2α,

且 -1≤ sin 2α ≤1,

所以 0≤ (sin α +cos α)^2 ≤2,

所以 -√2 ≤ sin α +cos α ≤ √2,

即 sin α +cos α 的最小值为 -√2,最大值为 √2.