a=(cosu,sinu),b=(cosv,sinv)

cos(u-v)=a∙b=(cosu,sinu)∙(cosv,sinv)=cosu cosv +sinu sinv------------------(1)

2.

a=(cosu,sinu),b=(cos(-v),sin(-v))=(cosv,-sinv)

cos(u+v)=a∙b=(cosu,sinu)∙(cosv,-sinv)=cosu cosv -sinu sinv--------------------(2)

3.a=(cosu,sinu),b=(cosv,sinv)

sin(u-v)=(axb) (在一個(gè)平面上)=cosu sinv - cosv sinu-------------------------(3)

4. a=(cosu,sinu),b=(cos(-v),sin(-v))=(cosv,-sinv)

sin(u+v)=ax(-b)(在一個(gè)平面上)=cosu sinv +cosv sinu------------------(4)

5.tan(u+v)=sin(u+v)/cos(u+v)=[tanu+tanv]/[1-tanu tanv]   (從(2),(4))

6. tan(u-v)   (從(1),(3))

7.設(shè) u=v, 從(4),sin2u=2sinu cosu

8. 設(shè) u=v, 從(2), cos2u= (cosu)^2-(sinu)^2----------(5)

9.設(shè)x=2u, 從(5), cosx=(cosx/2)^2-(sinx/2)^2-------------(6)

cosx=1-2(sinx/2)^2,

sin(x/2)=[(1-cosx)/2]^(1/2)

10. 從(6) cosx=(cosx/2)^2-1+(cosx/2)^2

cos(x/2)=[(1+cosx)/2]^(1/2)