A.maths:sine cosine

suppose acosx+bsinx+c=0
an equation of x has 2 roots m and n in the range of x is between 0 to 180

(a) show that a(cosm-cosn)+b(sinm- sinn)=0

(b)hence show that tan[(m+n)/2 = b/a
sol
(a) substitute m and n into the equation
acosm+bsinm+c=0...(1)
acosn+bsinn+c=0...(2)
(1)-(2)
a(cosm-cosn)+b(sinm- sinn)=0
(b)
Since
a(cosm-cosn)+b(sinm- sinn)=0
a(cosm-cosn)=-b(sinm- sinn)
(cosm-cosn)/(sinm- sinn)=-b/a
{-2sin[(m+n)/2]sin[(m-n)/2]}/{2cos[(m+n)/2]sin[(m-n)/2]}=-b/a
-sin[(m+n)/2]/cos[(m+n)/2]=-b/a
tan[(m+n)/2] = b/a

2007-04-09 22:08:43 補充：
sum to product formula(cosm-cosn)=-2sin[(m+n)/2]sin[(m-n)/2](sinm- sinn)=2cos[(m+n)/2]sin[(m-n)/2]