write
3 cos (ωt + 2) + 4 sin (ωt + 1)
as A cos (ωt + φ).
Find
A = ? φ= ?
To write the given expression in the form A cos (ωt + φ), we need to convert it into a single trigonometric function using trigonometric identities. Let's start by expanding the given expression:
3 cos (ωt + 2) + 4 sin (ωt + 1)
Using the trigonometric identity cos (a + b) = cos(a)cos(b) - sin(a)sin(b), we can rewrite the expression as:
3 [cos(ωt)cos(2) - sin(ωt)sin(2)] + 4 [sin(ωt)cos(1) + cos(ωt)sin(1)]
Simplifying further, we get:
3 cos(ωt)cos(2) - 3 sin(ωt)sin(2) + 4 sin(ωt)cos(1) + 4 cos(ωt)sin(1)
Using the trigonometric identities cos(2) = cos^2(1) - sin^2(1) and sin(2) = 2sin(1)cos(1), the expression becomes:
3 cos(ωt)[cos^2(1) - sin^2(1)] - 3 sin(ωt)[2sin(1)cos(1)] + 4 sin(ωt)cos(1) + 4 cos(ωt)sin(1)
Expanding it further, we have:
3 cos(ωt)cos^2(1) - 3 cos(ωt)sin^2(1) - 6 sin(ωt)sin(1)cos(1) + 4 sin(ωt)cos(1) + 4 cos(ωt)sin(1)
Rearranging terms, we get:
(3cos^2(1) + 4sin(1)cos(1))cos(ωt) + (4sin(1)cos(1) - 3sin^2(1) - 6sin(1)cos(1))sin(ωt)
Now, let's simplify the coefficients:
Coefficient of cos(ωt) = 3cos^2(1) + 4sin(1)cos(1)
Coefficient of sin(ωt) = 4sin(1)cos(1) - 3sin^2(1) - 6sin(1)cos(1)
Simplifying the coefficients gives:
Coefficient of cos(ωt) = 3cos^2(1) + 4sin(1)cos(1) = A cos(φ)
Coefficient of sin(ωt) = 4sin(1)cos(1) - 3sin^2(1) - 6sin(1)cos(1) = A sin(φ)
Now we can compare the coefficients to the form A cos (ωt + φ):
A cos(φ) = 3cos^2(1) + 4sin(1)cos(1)
A sin(φ) = 4sin(1)cos(1) - 3sin^2(1) - 6sin(1)cos(1)
To find the values of A and φ, we can use the properties of cos and sin functions. We can rewrite A cos(φ) and A sin(φ) in terms of A and φ while eliminating cos(1) and sin(1) as variables. By solving these equations simultaneously, A and φ can be determined.