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

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1 Answer
Aug 25, 2016

sin(alpha+beta)=sin(alpha)cos(beta)+cos(alpha)sin(beta)

Explanation:

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

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Proceeding, note that from the right triangle triangleACB, we have
sin(alpha+beta) = (AC)/(AB) = (AC)/1=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 triangleBDE, we have
sin(alpha) = (ED)/(BE) = (ED)/cos(beta)

=> ED = sin(alpha)cos(beta)

=> FC = ED = sin(alpha)cos(beta)


Next, as the sum of the angle of triangleBDE is pi, we have angleBED = pi-pi/2-alpha = pi/2-alpha.

As angleAEB=pi/2, this gives angleAED = pi/2+(pi/2-alpha)=pi-alpha.

Then, as angleFED = pi/2 by construction, we have angleAEF = pi-alpha-pi/2 = pi/2-alpha.

Looking at the right triangle triangleAFE, then, we can calculate angleFAE as
angleFAE = pi-pi/2-(pi/2-alpha) = alpha.

Calculating the cosine of that angle, we obtain
cos(alpha) = (AF)/(AE) = (AF)/sin(beta)

=> AF = cos(alpha)sin(beta)


Putting these together, we get our final result:

sin(alpha+beta) = AC

=FC+AF

=sin(alpha)cos(beta)+cos(alpha)sin(beta)