Help? It's in my calc homework packet

Question

Solve f'(x)=-3

(note this is the derivative of f(x)) when f(x)=7cosx-4sinx for the smallest positive x value.

Answer 1

f'(x) = -7 sin x - 4 cos x

(-7)^2 + (-4)^2 = 65

Thus, we may rewrite this as -√65 (7/√65 sin x + 4/√65 cos x) = -√65 sin(x + sin^-1(4/√65))

Then, -√65 sin(x + sin^-1(4/√65)) = -3, so sin(x + sin^-1(4/√65)) = 3/√65

The obvious answer for x is sin^-1(3/√65) - sin^-1(4/√65). However, this will be negative.

Thus, we need x + sin^-1(4/√65) = π - sin^-1(3/√65) (the next value to yield the correct value)

Then, x = π - sin^-1(3/√65) - sin^-1(4/√65)

x = 2.24117155145949 =128.40967106341°