Understanding the Sum and Difference Identities | Deriving and Applying the Trigonometric Identity for Sine

Sum and Difference Identitiessin(a-b) =

The sum and difference identities are trigonometric identities that involve the sine and cosine functions

The sum and difference identities are trigonometric identities that involve the sine and cosine functions. The specific identity you mentioned is:

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

This identity helps us find the sine of the difference of two angles (a and b) in terms of their sines and cosines.

To understand and prove this identity, we can use the angle addition identities. The sum and difference identities for sine and cosine are derived from these angle addition identities.

The angle addition identities are as follows:
1. sin(a + b) = sin a cos b + cos a sin b
2. cos(a + b) = cos a cos b – sin a sin b

Using these identities, we can derive the sum and difference identities:

1. To find sin(a – b), we can rewrite it as sin(a + (-b)).
Using the angle addition identity for sine, we have:
sin(a + (-b)) = sin a cos(-b) + cos a sin(-b)

2. We know that cos(-b) = cos b and sin(-b) = -sin b (due to the properties of even and odd functions).
Substituting these values, the equation becomes:
sin(a – b) = sin a cos b – cos a sin b

Therefore, sin(a – b) = sin a cos b – cos a sin b is the sum and difference identity for sine.

This identity is useful in trigonometry when we need to find the sine of the difference of two angles without calculating the individual sines of the angles separately.

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