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

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

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Answer 1 · 2016-08-25T20:56:37

$sin (α + β) = sin (α) cos (β) + cos (α) sin (β)$ $sin(alpha+beta)=sin(alpha)cos(beta)+cos(alpha)sin(beta)$

Explanation:

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

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Proceeding, note that from the right triangle $△ A C B$ $triangleACB$ , we have
$sin (α + β) = \frac{A C}{A B} = \frac{A C}{1} = A C$ $sin(alpha+beta) = (AC)/(AB) = (AC)/1=AC$

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

From the right triangle $△ B D E$ $triangleBDE$ , we have
$sin (α) = \frac{E D}{B E} = \frac{E D}{cos (β)}$ $sin(alpha) = (ED)/(BE) = (ED)/cos(beta)$

$\Rightarrow E D = sin (α) cos (β)$ $=> ED = sin(alpha)cos(beta)$

$\Rightarrow F C = E D = sin (α) cos (β)$ $=> FC = ED = sin(alpha)cos(beta)$

Next, as the sum of the angle of $△ B D E$ $triangleBDE$ is $π$ $pi$ , we have $∠ B E D = π - \frac{π}{2} - α = \frac{π}{2} - α$ $angleBED = pi-pi/2-alpha = pi/2-alpha$ .

As $∠ A E B = \frac{π}{2}$ $angleAEB=pi/2$ , this gives $∠ A E D = \frac{π}{2} + (\frac{π}{2} - α) = π - α$ $angleAED = pi/2+(pi/2-alpha)=pi-alpha$ .

Then, as $∠ F E D = \frac{π}{2}$ $angleFED = pi/2$ by construction, we have $∠ A E F = π - α - \frac{π}{2} = \frac{π}{2} - α$ $angleAEF = pi-alpha-pi/2 = pi/2-alpha$ .

Looking at the right triangle $△ A F E$ $triangleAFE$ , then, we can calculate $∠ F A E$ $angleFAE$ as
$∠ F A E = π - \frac{π}{2} - (\frac{π}{2} - α) = α$ $angleFAE = pi-pi/2-(pi/2-alpha) = alpha$ .

Calculating the cosine of that angle, we obtain
$cos (α) = \frac{A F}{A E} = \frac{A F}{sin (β)}$ $cos(alpha) = (AF)/(AE) = (AF)/sin(beta)$

$\Rightarrow A F = cos (α) sin (β)$ $=> AF = cos(alpha)sin(beta)$

Putting these together, we get our final result:

$sin (α + β) = A C$ $sin(alpha+beta) = AC$

$= F C + A F$ $=FC+AF$

$= sin (α) cos (β) + cos (α) sin (β)$ $=sin(alpha)cos(beta)+cos(alpha)sin(beta)$

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

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