Derive the sine sum formula using the geometrical construction given in the figure?

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1 Answer
Aug 25, 2016

sin(α+β)=sin(α)cos(β)+cos(α)sin(β)

Explanation:

First, construct two additional lines and label the intersections/vertices as follows:

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Proceeding, note that from the right triangle ACB, we have
sin(α+β)=ACAB=AC1=AC

Then, it remains to calculate AC. To do so, we will calculate AF and FC separately, and then add them together.


From the right triangle BDE, we have
sin(α)=EDBE=EDcos(β)

ED=sin(α)cos(β)

FC=ED=sin(α)cos(β)


Next, as the sum of the angle of BDE is π, we have BED=ππ2α=π2α.

As AEB=π2, this gives AED=π2+(π2α)=πα.

Then, as FED=π2 by construction, we have AEF=παπ2=π2α.

Looking at the right triangle AFE, then, we can calculate FAE as
FAE=ππ2(π2α)=α.

Calculating the cosine of that angle, we obtain
cos(α)=AFAE=AFsin(β)

AF=cos(α)sin(β)


Putting these together, we get our final result:

sin(α+β)=AC

=FC+AF

=sin(α)cos(β)+cos(α)sin(β)