Question

Question #0a796

Answer 1

`3.7 cos(pi t-5.953)` i.e. the constants are `A = 3.7`, `omega = pi` and `phi =2pi-tan^{-1}(12/35)=5.953`

Explanation:

To express the given function in the form `A cos (omega t + phi)`, notice that according to the trigonometric sum rule

`A cos (omega t + phi) = A [cos (omega t )cos (phi)-sin(omega t)sin(phi)] = -A sin(phi) sin(omega t) + A cos(phi)cos(omega t)`

Comparing with the expression given, we have `omega = pi` nd

`A sin(phi) = -1.2, qquad A cos(phi) = 3.5`

Thus

`A^2 = = (A sin(phi))^2 + (A cos(phi))^2= (-1.2)^2+(3.5)^2 = 13.69 = 3.7^2`

So, `A = 3.7`

Now,
`tan(phi) = {A sin(phi)}/{A cos(phi)} = -{1.2}/{3.5} = -0.3429`

Since `sin(phi)` is negative while `cos(phi)` is positive, the angle `phi` must be in the fourth quadrant. So

`phi =2pi-tan^{-1}(12/35)=5.953`