【答案】
分析:由于sin2x=2sinxcosx,設(shè)sinx+cosx=t,則sin2x=2sinxcosx=t
2-1,這樣一來(lái),原函數(shù)就轉(zhuǎn)化為關(guān)于t的二次函數(shù)了,
接下來(lái),求關(guān)于t的二次函數(shù)的值域.
解答:解:設(shè)sinx+cosx=t,則t∈
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214546323109654/SYS201310232145463231096021_DA/0.png)
∴sin2x=2sinxcosx=t
2-1,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214546323109654/SYS201310232145463231096021_DA/1.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214546323109654/SYS201310232145463231096021_DA/2.png)
∵t∈
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214546323109654/SYS201310232145463231096021_DA/3.png)
∴y∈
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214546323109654/SYS201310232145463231096021_DA/4.png)
∴值域:
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214546323109654/SYS201310232145463231096021_DA/5.png)
.
點(diǎn)評(píng):有關(guān)sinx+cosx與sinxcosx的三角函數(shù)式的值域問(wèn)題,通常采用換元法,設(shè)sinx+cosx=t,則sin2x=2sinxcosx=t
2-1.
轉(zhuǎn)化成關(guān)于新的變量t的函數(shù)值域問(wèn)題來(lái)解決,特別要注意新變量t的取值范圍.