将函数y=-2sin^2(x)+2根号3sinxcosx化成正弦函数 已知函数f(x)=2sin²x 2根号3sinxc...
\u5df2\u77e5\u51fd\u6570y=sin^2x+2\u6839\u53f73sinxcosx-cos^x,(x\u2208R\uff09\u3002y=(sinx)^2+2\u221a3*sinxcosx-(cosx)^2=\u221a3sin(2x)-cos(2x)=2sin(2x-\u03c0/6) \u3002
\uff081\uff09\u6700\u5c0f\u6b63\u5468\u671f\u4e3a T=2\u03c0/2=\u03c0 \u3002
\uff082\uff09\u7531 -\u03c0/2+2k\u03c0<=2x-\u03c0/6<=\u03c0/2+2k\u03c0 \uff0c\u5f97 -\u03c0/6+k\u03c0<=x<=\u03c0/3+k\u03c0 \uff0ck\u2208Z \uff0c
\u56e0\u6b64\u51fd\u6570\u7684\u5355\u8c03\u9012\u589e\u533a\u95f4\u662f [ -\u03c0/6+k\u03c0 \uff0c\u03c0/3+k\u03c0] \uff0ck\u2208Z \u3002
\uff083\uff09\u56e0\u4e3a 0<=x<=\u03c0/2 \uff0c\u6240\u4ee5 -\u03c0/6<=2x-\u03c0/6<=5\u03c0/6 \uff0c
\u7531\u6b63\u5f26\u51fd\u6570\u7684\u6027\u8d28\uff0c\u5f53 2x-\u03c0/6= -\u03c0/6 \u5373 x=0 \u65f6\uff0c\u51fd\u6570\u53d6\u6700\u5c0f\u503c -1 \uff0c
\u5f53 2x-\u03c0/6=\u03c0/2 \u5373 x=\u03c0/3 \u65f6\uff0c\u51fd\u6570\u53d6\u6700\u5927\u503c 2 \uff0c
\u56e0\u6b64\u51fd\u6570\u503c\u57df\u4e3a [-1 \uff0c2] \u3002
f(x)=2sin^2x+2\u221a3sinxcosx-1
=1-cos2x+\u221a3sin2x-1
=\u221a3sin2x-cos2x
=2sin(2x-\u03c0/6)
f(x)\u662f\u6b63\u5f26\u51fd\u6570\uff0c\u5176\u4e2d\u5fc3\u5bf9\u79f0\u70b9\u4f4d\u4e8e\u51fd\u6570\u4e0ex\u8f74\u7684\u4ea4\u70b9\u4e0a
\u6839\u636e\u9898\u610f\uff0cf(x)\u7684\u56fe\u50cf\u5173\u4e8e\u70b9(x,0)\u5bf9\u79f0
\u6240\u4ee52sin(2x-\u03c0/6)=0
2x-\u03c0/6=k\u03c0
x=k\u03c0/2+\u03c0/12\uff0c\u5176\u4e2dk\u662f\u4efb\u610f\u6574\u6570
sinxcosx=1/2sin2x
辅助角公式：
asinx+bcosx=√(a²+b²)sin(x+φ)
y=-2sin^2(x)+2√3sinxcosx
=- (1-cos2x)+√3sin2x
=√3sin2x+cos2x-1
=2(√3/2sin2x+1/2cos2x)-1
=2sin(2x+π/6)-1
下面可以研究该函数的图像与性质
中学生数理化团队为你答疑！
f(x)=2cos²x+2根号3sinxcosx=2-2sin²x+2根号3sinxcosx
若f(C)=2，有-2sin²C+2根号3sinCcosC=0，在三角形ABC中sinC≠0，2sinC=2根号3cosC
即tanC=根号3,得C=π/3.
三角形ABC面积=1/2absinC=1/4根号3*ab≤1/16根号3*(a+b)²=根号3
当a=b=2时，三角形ABC面积最大值为根号3。
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