# TrigonometryAmplitude, Period, frequency and phase angle

#### jenney

##### New member
HELP!!!!

totally lost and confused with this question:
A machine is subject to two vibrations at the same time.
one vibration has the form: 2cosωt and the other vibration has the form: 3 cos(ωt+0.785). (0.785 is actually expressed as pi/4)
determine the resulting vibration and express it in the general form of: n cos(ωt±α)

#### skeeter

##### Well-known member
MHB Math Helper
HELP!!!!

totally lost and confused with this question:
A machine is subject to two vibrations at the same time.
one vibration has the form: 2cosωt and the other vibration has the form: 3 cos(ωt+0.785). (0.785 is actually expressed as pi/4)
determine the resulting vibration and express it in the general form of: n cos(ωt±α)
$2\cos(\omega t) + 3\cos \left(\omega t + \dfrac{\pi}{4} \right)$

$2\cos(\omega t) + 3\left[\cos(\omega t)\cos\left(\dfrac{\pi}{4}\right) - \sin(\omega t)\sin\left(\dfrac{\pi}{4}\right)\right]$

$2\cos(\omega t) + \dfrac{3\sqrt{2}}{2}\left[\cos(\omega t) - \sin(\omega t)\right]$

$\dfrac{4+3\sqrt{2}}{2}\cos(\omega t) - \dfrac{3\sqrt{2}}{2}\sin(\omega t)$

note $A\cos{x} + B\sin{x} = R\cos(x - \alpha)$, where ...

$R = \sqrt{A^2+B^2}$ and $\alpha = \arctan\left(\dfrac{B}{A}\right)$

... see what you can do from here. Note that the values for $R$ and $\alpha$ are not "nice".