Question

已知θ为锐角,用三角函数线求证1<sinθ+cosθ≤根号2

（1）1<sinθ+cosθ≤根号2
（2）sinθ＜θ＜tanθ

Answer 1

y=sinθ+cosθ
y²=sin²θ+cos²θ+2sinθ*cosθ=1+sin2θ
0＜sin2θ≤1
1＜罩余y²≤2
∴1<sinθ+cosθ≤√2

2
由图可知
Sinα＝ab/游仔r ab弦
α＝ab/r ab弧
tan α=ac/r ac切线
∵ac＞ab＞ab
∴sinα＜α＜tanα

图片传不物磨滚上!!

Answer 2

sinθ+cosθ= 根号2 * sin（θ+π/4) θ∈(0,π/租亮销2) → sin（θ+π/4) ∈(2分之根号2, 1]
∴ sinθ+cosθ 属于弊游 (1,根号2]

设f(x) = sinx - x g(x) = tanx - x x属于(0,π/2)
f'(x) = cosx - 1 <= 0 f(0) = 0 可知 sinx < x
g'(x) = 1+ tanx * tanx - 1 = tanx * tanx 当x为锐角时候 g'(x) > 0 又 g(0) = 0 可知 x < tanx
sinθ＜θ＜tanθ得证键掘

Answer 3

1 都平方一下，就出来了