The Sum-To-Product Formula For Sin And The Unit Circle Concept

sin(A+B)

sinAcosB+cosAsinB

The formula for sin(A + B) is:

sin(A + B) = sin(A) cos(B) + cos(A) sin(B)

This formula is also known as the sum-to-product formula for sin.

To understand this formula, we need to use the concept of the unit-circle. Consider a point P on the unit circle with an angle A from the positive x-axis and another point Q on the same circle with an angle B from the positive x-axis. Then, the point R obtained by adding the angles A and B will have coordinates (cos(A + B), sin(A + B)).

Now, consider the right-angled triangle OAP, where O is the origin, AP is the perpendicular drawn from P to the x-axis, and OA is the radius of the circle (which is equal to 1). By definition, sin(A) is equal to the y-coordinate of the point P, which is AP/OA. Similarly, cos(A) is equal to the x-coordinate of P, which is OP/OA. Similarly, we can define sin(B) and cos(B) using the triangle OQB.

Then, using the geometry of the unit circle and trigonometric ratios, we can show that the y-coordinate of the point that results from adding the angles A and B is given by:

sin(A + B) = sin(A) cos(B) + cos(A) sin(B)

which is the formula we started with.

So, sin(A + B) is a combination of sin(A), cos(B), cos(A), and sin(B) and depends on the values of A and B.

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