Using high school geometry only, show

that sin(A+B) = sinAcosB + sinBcosA

how would i show how one side equals the other side??(Doh)(Shake)(Shake)(Worried)(Surprised)(Wond ering)

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- Sep 25th 2009, 11:59 AMSneakyproving an addition identity????
Using high school geometry only, show

that sin(A+B) = sinAcosB + sinBcosA

how would i show how one side equals the other side??(Doh)(Shake)(Shake)(Worried)(Surprised)(Wond ering) - Sep 25th 2009, 01:28 PMKrizalid
consider

http://img243.imageshack.us/img243/9134/recqe9.png

in this figure, we have that NOPR is a rectangle, ON was extended and we get the point M. Join M and P and extend NR, point Q borns. Finally make QP orthogonal to MP and join M and Q.

let $\displaystyle \measuredangle\,OMP=\alpha$ and $\displaystyle \measuredangle\,QMP=\beta.$ It's easy to prove that $\displaystyle \measuredangle\,OMP=\measuredangle\,PQR=\alpha.$

compute $\displaystyle \sin(\alpha+\beta)$ and we have it's equal to $\displaystyle \frac{{\overline {QN} }}{{\overline {QM} }} = \frac{{\overline {QR} + \overline {RN} }}{{\overline {QM} }} = \frac{{\overline {QR} }}{{\overline {QM} }} + \frac{{\overline {OP} }}{{\overline {QM} }}.$ Multiply the first quotient top and bottom by $\displaystyle \overline{QP}$ and in the same fashion for the second quotient by $\displaystyle \overline{MP}$ and you'll get the identity.