productToSumAndSumToProduct

Product to sum to product §

• given $\sin (A+B)=\sin A\cos B+\cos A\sin B$ $\sin (A-B)=\sin A\cos B-\cos A\sin B$
• adding these gives:$\sin (A+B)+\sin (A-B)=2\sin A\cos B$
• and so$\sin A\cos B=\frac{1}{2}(\sin (A+B)+\sin (A-B))$
• or writing $A+B=P$ and $A-B=Q$ $\sin P+\sin Q=2\sin \left(\frac{P+Q}{2}\right)\cos \left(\frac{P-Q}{2}\right)$
• these are some of 8 formulas D:, for sinAcosB, cosAsinB, cosAcosB etc etc.. these are all obtained by arranging the stuff in a specific way.
• p 195-196 textbook for derivation of these (only the product of sum identities are in the formula booklet)
  • you need to solve for the sum to product ones yourself :D

example question §

• solve $4\sin 5x\sin 3x+2\cos 8x=1$
• prove that $\frac{\sin 7x-\sin 3x}{\cos 6x+\cos 4x}$
  • need to derive the corresponding sum to product formulas.