Sin Sum Identity: The Trigonometric Sum-To-Product Identity And Its Proof.

Sin Sum Identity

sin(A+B)=sinAcosB+cosAsinB

The sin sum identity, also known as the sum-to-product identity, is a trigonometric identity that expresses the sine of the sum of two angles in terms of the sines and cosines of the individual angles. The identity is given as:

sin(α + β) = sin α cos β + cos α sin β

where α and β are any two angles.

This identity can be proved geometrically using the figure below:


In the figure, O is the origin, A is a point on the unit circle, and P and Q are two points on the line passing through the origin and A, making angles of α and β with the positive x-axis, respectively. The coordinates of A, P, and Q are given by:

A = (cos(α + β), sin(α + β))
P = (cos α, sin α)
Q = (cos β, sin β)

By the distance formula, we have:

OP = cos α
OQ = cos β
AP = cos(α + β)

Using the Pythagorean theorem, we also have:

OA = 1
AP^2 = OA^2 – OP^2
AQ^2 = OA^2 – OQ^2

Substituting these expressions and simplifying, we can obtain:

2AP·AQ = 2cos α·cos β + sin α·sin β

Dividing both sides by 2 sin α·sin β, we get:

sin(α + β) = sin α cos β + cos α sin β

which is the desired identity.

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