Question

在三角形ABC中,已知三个内角A.B.C成等差数列,则tanA/2+tanC/2+根号3倍的tanA/2乘以tanC/2

Answer 1

A、B、C成等差数列,则2B=A+C

A+B+C=180°

3B=180°

B=60°

tan(A/2+C/2)=[tan(A/2)+tan(C/2)]/[1-tan(A/2)tan(C/2)]

[tan(A/2)+tan(C/2)]/[1-tan(A/2)tan(C/2)]

=tan(A/2+C/2)

=tan[(180°-B)/2]

=tan[(180°-60°)/2]

=tan60°

=√3

tan(A/2)+tan(C/2)=√3-√3tan(A/2)tan(C/2)

tan(A/2)+tan(C/2)+√3tan(A/2)tan(C/2)=√3