The identities cos(a-b)=cos(a)cos(b)sin(a)sin(b) and sin(a-b)=sin(a)cos(b)-cos(a)sin(b) are occasionally useful. Justify them. One method is to use rotation matricies. Another method is to use the established identities for cos(a+b) and sin (a+b).
Sounds like a good justification to me. Oh, did you mean prove them? In that case, using the identities,
cos(a-b) = cos(a + (-b)) = cos(a) cos(-b) - sin(a) sin(-b)
= cos(a)cos(b) + sin(a) sin(b)
sin(a-b) = sin(a + (-b)) = sin(a) cos(-b) + cos(a) sin(-b)
= sin(a) cos(b) - cos(a) sin(b)
Sin2x
To justify the identities cos(a-b) = cos(a)cos(b)sin(a)sin(b) and sin(a-b) = sin(a)cos(b) - cos(a)sin(b), we can utilize both rotation matrices and established identities for cos(a+b) and sin(a+b).
Method 1: Using Rotation Matrices
Rotation matrices are a convenient tool to understand trigonometric identities. Let's consider a point P in a 2D coordinate system with coordinates (x, y). If we rotate this point counterclockwise by an angle θ, the new coordinates of the point P', denoted as (x', y'), can be expressed by the following equations:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Now, let's assume we have a point P located at (cos(a), sin(a)). To rotate this point by an angle b, we substitute x = cos(a), y = sin(a), and θ = -b into the rotation matrix equations:
x' = cos(a) * cos(-b) - sin(a) * sin(-b)
y' = cos(a) * sin(-b) + sin(a) * cos(-b)
Using the trigonometric identities cos(-θ) = cos(θ) and sin(-θ) = -sin(θ), the equations become:
x' = cos(a) * cos(b) + sin(a) * sin(b) = cos(a + b)
y' = sin(a) * cos(b) - cos(a) * sin(b) = sin(a + b)
Hence, we have shown that cos(a-b) = cos(a)cos(b)sin(a)sin(b) and sin(a-b) = sin(a)cos(b) - cos(a)sin(b) using rotation matrices.
Method 2: Using Established Identities for cos(a+b) and sin(a+b)
We can also use the well-established identities for cos(a+b) and sin(a+b) to justify the given identities. The identities for cos(a+b) and sin(a+b) are:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
Let's apply these identities by assuming b = -b:
cos(a - b) = cos(a + (-b)) = cos(a)cos(-b) - sin(a)sin(-b)
sin(a - b) = sin(a + (-b)) = sin(a)cos(-b) + cos(a)sin(-b)
Using cos(-θ) = cos(θ) and sin(-θ) = -sin(θ), the equations become:
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
Therefore, we have justified the identities cos(a-b) = cos(a)cos(b)sin(a)sin(b) and sin(a-b) = sin(a)cos(b) - cos(a)sin(b) using the established identities for cos(a+b) and sin(a+b).
Both methods provide valid justifications for the given identities.