Question

在三角形ABC中，AB=根号2+根号6，C=30度，则AC+AB的最大值

则AC+BC的最大值，上面的打错啦

Answer 1

AB=√2+√6
C=30°
A+B+C=180°
A+B=150°
B=150°-A
0＜A，B＜150°
AC/sinB=BC/sinA=AB/sinC
AC=ABsinB/sinC，BC=ABsinA/sinC
AC+BC=AB/sinC *(sinA+sinB) = (√2+√6)/2 {sinA+sin(150°-A)}
= (√2+√6)/2 {sinA+sin150°cosA-cos150°sinA}
= (√2+√6)/2 {sinA+1/2cosA+√3/2sinA}
= (√2+√6)/4 {(2+√3)sinA+cosA}
令cost=(2+√3)/(8+4√3)，sint=1/(8+4√3)
= (√2+√6)/4*(8+4√3) {sinAcost+cosAsint}
= (2+3√6+3√2)sin(A+t)
当A+t=90°时，AC+BC的最大值为2+3√6+3√2