The Sin Difference Identity: A Trigonometric Formula Explained

Sin Difference Identity

sin(A-B)=sinAcosB-cosAsinB

The Sin Difference Identity, also known as the Angle Sum or Subtraction Formula for Sine, is a trigonometric identity that relates the difference between two angles to sine functions. It can be expressed in different forms, but one of the most common ones is:

sin(a – b) = sin(a)cos(b) – cos(a)sin(b)

This identity states that the sine of the difference between two angles (a and b) is equal to the sine of the first angle (a) times the cosine of the second angle (b) minus the cosine of the first angle (a) times the sine of the second angle (b).

To understand why this identity works, we need to use the definitions of sine and cosine in terms of a unit circle. When we draw an angle in standard position (with its vertex at the origin, its initial side along the positive x-axis, and its terminal side rotating counterclockwise), it intersects the unit circle at a point (x,y) where x = cos(theta) and y = sin(theta), with theta being the angle.

Using this information, we can see that sin(a – b) is the y-value of the point where the line passing through angles a – b and the positive x-axis intersects the unit circle. We can express this y-value in terms of the coordinates of two other points on the unit circle: the one corresponding to a (cos(a), sin(a)) and the one corresponding to b (cos(b), sin(b)).

To do this, we use the fact that the sine of an angle is the y-coordinate of the corresponding point on the unit circle. We can then apply the angle addition formula for sine, which states that sin(x + y) = sin(x)cos(y) + cos(x)sin(y) for any angles x and y. In our case, we have:

sin(a – b) = sin(a)cos(-b) + cos(a)sin(-b) (using the angle addition formula with -b instead of b)
sin(a – b) = sin(a)cos(b) – cos(a)sin(b) (using the symmetry of sine and cosine)

Therefore, we get the Sin Difference Identity.

This identity has many applications in trigonometry, such as finding exact values of trigonometric functions, proving other identities, and solving equations involving trigonometric functions.

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