- #1
Vital
- 108
- 4
Homework Statement
Hello!
Please, take a look at the following equation and help me to understand where the authors got the value of φ = π/3. I don't see where it is derived from as no additional conditions are given.
Homework Equations
x(t) = 5e(-t/5) cos(t) + 5e(-t/5) √3 sin(t)
The Attempt at a Solution
Here is how I proceed:
x(t) = 5e(-t/5) (cos(t) + √3 sin(t))
Now, given the formula:
f(x) = a sin(wx) + b cos(wx) + B (w > 0) is the same as
f(x) = √a2+b2 sin(wx + φ) + B
In my case, B = 0, so I rewrite the expression as:
x(t) = 5e(-t/5) ( √3 sin(t) + cos(t) )
x(t) = 5e(-t/5) √√32+12 sin(t + φ)
x(t) = 10e(-t/5) sin(t + φ)
But in the book they have:
x(t) = 10e(-t/5) sin(t + π/3)
Where did they get φ = π/3 from?
Thank you!