有关三角函数的最值问题在△ABC中,A、B、C分别为三角形内角,a、b、c为其所对边,已知2√2*(sin^2A-sin^2C)=(a-b)sinB,△ABC外接圆半径为√2.1、求周长范围 2、求a²+b²的范围
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![有关三角函数的最值问题在△ABC中,A、B、C分别为三角形内角,a、b、c为其所对边,已知2√2*(sin^2A-sin^2C)=(a-b)sinB,△ABC外接圆半径为√2.1、求周长范围 2、求a²+b²的范围](/uploads/image/z/4333170-66-0.jpg?t=%E6%9C%89%E5%85%B3%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%E7%9A%84%E6%9C%80%E5%80%BC%E9%97%AE%E9%A2%98%E5%9C%A8%E2%96%B3ABC%E4%B8%AD%2CA%E3%80%81B%E3%80%81C%E5%88%86%E5%88%AB%E4%B8%BA%E4%B8%89%E8%A7%92%E5%BD%A2%E5%86%85%E8%A7%92%2Ca%E3%80%81b%E3%80%81c%E4%B8%BA%E5%85%B6%E6%89%80%E5%AF%B9%E8%BE%B9%2C%E5%B7%B2%E7%9F%A52%E2%88%9A2%2A%28sin%5E2A-sin%5E2C%29%3D%28a-b%29sinB%2C%E2%96%B3ABC%E5%A4%96%E6%8E%A5%E5%9C%86%E5%8D%8A%E5%BE%84%E4%B8%BA%E2%88%9A2.1%E3%80%81%E6%B1%82%E5%91%A8%E9%95%BF%E8%8C%83%E5%9B%B4++++++++++2%E3%80%81%E6%B1%82a%26%23178%3B%2Bb%26%23178%3B%E7%9A%84%E8%8C%83%E5%9B%B4)
有关三角函数的最值问题在△ABC中,A、B、C分别为三角形内角,a、b、c为其所对边,已知2√2*(sin^2A-sin^2C)=(a-b)sinB,△ABC外接圆半径为√2.1、求周长范围 2、求a²+b²的范围
有关三角函数的最值问题
在△ABC中,A、B、C分别为三角形内角,a、b、c为其所对边,已知2√2*(sin^2A-sin^2C)=(a-b)sinB,△ABC外接圆半径为√2.
1、求周长范围 2、求a²+b²的范围
有关三角函数的最值问题在△ABC中,A、B、C分别为三角形内角,a、b、c为其所对边,已知2√2*(sin^2A-sin^2C)=(a-b)sinB,△ABC外接圆半径为√2.1、求周长范围 2、求a²+b²的范围
(1)根据正弦定理 ∵△ABC外接圆半径为√2.
∴a/sinA=b/sinB=c/sinC=2√2.
又2√2*(sin^2A-sin^2C)=(a-b)sinB
即2√2*(a²/8-c²/8)=(a-b)√2b/4
即a²-c²=ab-b²
b²+a²-c²=ab
cosC=1/2
∠C=60°
a+b+c
=2√2(sinA+sinB+sinC)
=2√2(sinA+sin(A+60°)+√3/2)
=2√2(sinA+sinA/2+√3cosA/2+√3/2)
=2√2(3*sinA/2+√3cosA/2+√3/2)
=2√2[√3(√3sinA/2+cosA/2)+√3/2]
=2√2(√3*sin(A+30°)+√3/2)
∵A属于(0,2π/3)
A+π/6属于(π/6,5π/6)
sin(A+π/6)属于(1/2,1]
∴2√2(√3*sin(A+30°)+√3/2)属于(2√6,3√6].
(2)a²+b²
=2√2((sinA)²+sin(A+π/6)²)
=2√2((sinA)²+(sinA/2+√3cosA/2)²)
=2√2(5sinA/4+√3sinA*cosA/2+3(1-(sinA)²)/4)
=2√2((sinA)²/2+√3sin2A/4)
=2√2(1-cos2A+√3sin2A)/4
=√2(1+2sin(2A-π/6))/2
可得sin(2A-π/6)属于(-1/2,1]
a²+b²属于(0,3√2/2]