为什么sinX+cosX的值域是[负根号2,根号2]?
来源:学生作业帮助网 编辑:作业帮 时间:2024/08/14 09:35:22
![为什么sinX+cosX的值域是[负根号2,根号2]?](/uploads/image/z/1699371-27-1.jpg?t=%E4%B8%BA%E4%BB%80%E4%B9%88sinX%2BcosX%E7%9A%84%E5%80%BC%E5%9F%9F%E6%98%AF%5B%E8%B4%9F%E6%A0%B9%E5%8F%B72%2C%E6%A0%B9%E5%8F%B72%5D%3F)
xSj@YFMKoEU7V&PJ`M
qtcAk}D
C#AHH2s\ʹӹݙQ>/\C݆嵆z3h~dx_8O/?剥}PQP4Fظ"ߺ%duđi)bnLY{5]x"gxIsoG |X.sYebl&8bNSW4s; ވ9D#_ϝ(elW8X^j/
{Ҧ
['w7I!8{<$+v7o(2C;ᩁtTF-".}9;v Mccw䑔@|yAT6,h@]7#@1FhN?4s_R(
为什么sinX+cosX的值域是[负根号2,根号2]?
为什么sinX+cosX的值域是[负根号2,根号2]?
为什么sinX+cosX的值域是[负根号2,根号2]?
令 y=sinx+cosx
则 y=√2(sinxcosπ/4+cosxsinπ/4)=√2sin(x+π/4)
因为 -1≤sin(x+π/4)≤1
所以,-√2≤y≤√2
即 -√2≤sinx+cosx≤√2
y=sinx+cosx
=√2( √2/2sinx+√2/2cosx)
=√2sin(x+π/4)
所以有sinX+cosX的值域是[负根号2,根号2]。
注:sin(π/4)=cos(π/4)=√2/2
可以化为根号2乘sin(x+45°)
这个应该明白了吧
sinX+cosX=√2(√2/2*sinx+√2/2cosx)
=√2(cosπ/4*sinx+sinπ/4*cosx)
=√2sin(x+π/4) -1<=sin(x+π/4)<=1
所以sinX+cosX的值域是[负根号2,根号2]
sinX+cosX
= √2(sinXcosπ/4+cosXsinπ/4)
= √2 sin(X+π/4)
-1≤sin(X+π/4)≤1
-√2≤ √2sin(X+π/4)≤√2
∵ sinx+cosx=√2sin(x+π/4)
又,|sin(x+π/4)|≤1,即-1≤sin(x+π/4)≤1.
当sin(x+π/4)=-1时,原式=-√2;
当sin(x+π/4)=1时, 原式=√2。
∴sinx+cosx的值域时[-√2,√2].