高一数学三角函数题! 求详细过程!! 高一数学三角函数选择题求解题详细过程,高手请进!
\u9ad8\u4e00\u6570\u5b66\u4e09\u89d2\u51fd\u6570\uff01\u6025\u6c42\uff01\u8be6\u7ec6\u8fc7\u7a0b\uff0118\u9898\uff01\uff01\u4ee4t=sinx
y=f(x)=(a-b)(sinx)^2+2asinx+b
=(a-b)t^2+2at+b=g(t)
|t|<=1
a\u2260b
g(1)=a-b+2a+b=3a
g(-1)=a-b-2a+b=-a
g(1)*g(-1)=-3a^2<0
g(t)=0\uff0ct\u5728[-1\uff0c1]\u4e2d\u6709\u4e14\u53ea\u6709\u4e00\u4e2a\u89e3
\u6545y=f(x)\u4e0ex\u8f74\u6709\u4e24\u4e2a\u4ea4\u70b9
x=\u03c0/6\uff0csinx=1/2
\u9876\u70b9\uff08-1/(a-b)\uff0c4(ab-b^2+1)/4a\uff09
a-b=-2\uff0cab-b^2+1=7a
a(a+2)-(a+2)^2+1=7a
9a=-3\uff0ca=-1/3\uff0cb=5/3
\u8003\u8bd5\u4e0d\u80fd\u9760\u8fd9\u4e2a\u4f5c\u5f0a\uff01\uff01
\u7531 f(x)=0 \u5f97\u96f6\u70b9\u5750\u6807\u65b9\u7a0b \u33d2{2}x1=(1/2)^x1\uff0c\u2234 1<x1<\u221a2\uff1b
\uff08x\u221a2 \u65b9\u7a0b\u53f3\u7aef\u51fd\u6570\u5c0f\u4e8e 1/2\uff0c\u6b64\u65f6\u5de6\u7aef>1/2\uff0c\u65b9\u7a0b\u4e5f\u65e0\u6cd5\u6210\u7acb\uff09\uff1b
\u7531 g(x)=0 \u5f97\u96f6\u70b9\u5750\u6807\u65b9\u7a0b \u33d2{1/2}x2=(1/2)^x2\uff0c\u2234 1/\u221a2<x2<1/2\uff1b
\uff08x=1/2 \u4ea4\u70b9\u65b9\u7a0b\u5de6\u7aef\u51fd\u6570\u503c=1\uff0c\u53f3\u7aef\u51fd\u6570\u662f1/\u221a2\uff0c\u5de6>\u53f3\uff1bx=1/\u221a2 \u5de6\u7aef\u51fd\u6570\u503c\u662f1/2\uff0c\u53f3\u7aef\u51fd\u6570>1/2\uff0c\u5de6\u7aef<\u53f3\u7aef\uff09\uff1b
\u2234 1/\u221a2<x1*x2<\u221a2/2\uff1b
\u9009 A\uff1b
a/cosA=b/cosB,由正弦定理,
sinAcosB=cosAsinB,
sin(A-B)=0,-π<A-B<π,
A-B=0,A=B.
A+B=π-C,2A=π-C,cos2A=-cosC.
(1- cos2A)(2-cosC)=1+ cos2A+1,
(1+cosC)(2-cosC)=2-cosC,2-cosC>0,
1+cosC=1,cosC=0,C是三角形内角,C=π/2.
2.
设AB边上的高为x,则
S△ABC=1/2•x(x+√(4-x^2)), 0<x≤2.
设x=2sinα,0<α≤π/2.
S△ABC=2sin^2α+sin2α
=1-cos2α+sin2α
=1+√2sin(2α-π/4),-π/4<2α-π/4≤3π/4,-√2/2<sin(2α-π/4)≤1,
0<S≤1+√2.
亲,此小题的几何意义是:已知三角形的外接圆的π/4圆周角A所夹弦长AC为2,求三角形ABC面积的最大值。
可知当三角形ABC为等腰三角形时面积最大。
a/b=sinA/sinB ,a/b=cosA/cosB
sinA/sinB =cosA/cosB
sinB cosA=sinAcosB
sinB cosA-sinAcosB=0
sin(B-A)=0
A=B
又,(sinA)^2(2-cosC)=(cosB)^2+1/2
(sinA)^2[1+(1+cos2A)]=1-(sinA)^2+1/2
(sinA)^2[1+2(cosA)^2)]=3/2-(sinA)^2
(sinA)^2[3-(sinA)^2]=3/2-(sinA)^2
4(sinA)^4-8(sinA)^2+3=0
[2(sinA)^2-1][2(sinA)^2-3]=0
(sinA)^2=1/2 或(sinA)^2=3/2
sinA=√2/2 或 sinA=√6/2>1(舍去)
A=π/4
A=B=π/4
C=π/2
2)A=π/4 ,a=2
2R=b/sinB=c/sinC=a/sinA=2/√2/2=2√2
S=1/2bcsinA=1/2*(2R)^2sinBsinCsinπ/4
=1/2*(2√2)^2sinBsinC*√2/2
=2√2sinBsinC
=sin(3π/4-B)
=2√2sinB(√2/2cosB+√2/2sinB)
=2sinBcosB+2(sinB)^2
=sin2B+(1-cos2B)
=sin2B-cos2B+1
=√2sin(2B-π/4)+1
B在(0,3π/4)
2B-π/4在(-π/4,5π/4)
sin(2B-π/4在(-π/4,5π/4)值域为:这(-√2/2,1]
S=√2sin(2B-π/4)+1值域为:这(0,1+√2]
S的取值范围是:(0,1+√2]
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