因为：
(a+b)(cosAcosB-sinAsinB) = -c(cosA+cosB)
(a+b)cos(A+B) = -c(cosA+cosB)
(a+b)cos(180°-C)=-c(cosA+cosB)
-(a+b)cosC = -c(cosA+cosB)
(a+b)cosC = c(cosA+cosB)
又因为：
a/sinA = b/sinB = c/sinC = 2R 注：R 为该三角形外接圆的半径
所以上式可以转化为：
(2RsinA+2RsinB)cosC = 2RsinC(cosA+cosB)
(sinA+sinB)cosC = sinC(cosA+cosB)
2sin[(A+B)/2]*cos[(A-B)/2] * cosC = sinC * 2cos[(A+B)/2]cos[(A-B)/2]
sin[(180°-C)/2] * cosC = sinC * cos[(180°-C)/2]
sin(90°-C/2) * cosC = sinC * cos(90° -C/2)
cos(C/2)*cosC = sinC * sin(C/2)
cos(C/2)*cosC = [2sin(C/2)*cos(C/2)] * sin(C/2)
cosC = 2sin²(C/2)
因为 cosC = 1 - 2sin²(C/2)
所以，2sin²(C/2) = 1 - 2sin²(C/2)
那么：cosC = 2sin²(C/2) = 1/2
所以 C = 60° = π/3
因此：
sin(20C+π/4)
= sin(20π/3 + π/4)
= sin(6π + 2π/3 + π/4)
= sin(2π/3 + π/4)
=sin(2π/3)cos(π/4) +cos(2π/3)sin(π/4)
=√3/2 * √2/2 - 1/2 * √2/2
=(√6 - √2)/4
因为 S = 1/2*ab*sinC
则：
4S/b + bc
=2absinC/b + bc
=2asinC + bc
=2√3*a + b*√3
=√3 * (2a + b)
=√3 * (2*2RsinA + 2R*sinB) 注：2R = c/sinC =√3/(√3/2) = 2
=√3 * (4sinA + 2sinB)
=2√3 * (2sinA + sinB)
=2√3 * [2sin(120°-B) + sinB]
=2√3 * [2sin120°cosB - 2cos120°sinB + sinB]
=2√3 * [√3 * cosB + sinB + sinB]
=2√3 * [√3 cosB + 2sinB]
=2√3 * √7 * [√3/√7 * cosB + 2/√7 * sinB]
=2√21 * [sinαcosB + cosα*sinB] 注：sinα=√3/√7, α≈40.89°
=2√21 * sin(α+B)
可见，sinα < sin(α+B) ≤ sin90°
因此：
2√21 * (√3/√7) < 2√21 * sin(α+B) ≤ 2√21 * 1
6 < 2√21 * sin(α+B) ≤ 2√21
即 4S/b + bc 的取值范围为：(6, 2√21]
