Problems identifying harmonic motion

Question

Not sure why I am having so much trouble with this. I have a function f(t) = -cos(t) + 3sin(t-pi/6). I am trying to find the amplitude, period, and phase angle. But, I am under the impression that because the arguments are different in each trig function the function do not map simple harmonic motion. Is this right? If the arguments of my compound function matched I would use the identity, acos(t) + bsin(t) = Asin(wt + g) = Asin(wt)cos(g) + Acos(wt)sin(g), therefor since this is a linear combination, the coefficients must match. ... solved. If someone would be able to illuminate where I am misunderstanding it would be greatly appreciated. Or recommend a book that explains this. A free PDF works too. Thank you!!

Answer 1

One way to proceed is to simply write

$$\sin(t-\pi/6)=\cos(\pi/6)\sin t-\sin(\pi/6) \cos t$$

Then,

$$-\cos t+3\sin (t-\pi/6)=3\cos(\pi/6)\,\sin t-(3\sin(\pi/6)+1)\, \cos t$$

Finally,

$$\bbox[5px,border:2px solid #C0A000]{-\cos t+3\sin (t-\pi/6)=A\sin (t+\phi)}$$

where

$$\tan \phi=\frac{-5\sqrt{3}}{9}$$

and

$$A=\sqrt{13}$$

Answer 2

Try converting the problem into this form. See Dr. MV's answer to see how to do this (i.e. use sum of angle formulas). $$\cos\alpha\sin t + \sin\alpha\cos t$$ Quite often though we may have something of this form, for example $$3\sin t+4\cos t$$ The problem is that $$\sin^2\alpha + \cos^2\alpha =1$$ but $$3^2+4^2=25$$ So this requires us to do some manipulation, the point being to divide by the length of the hypotenuse of a right angle triangle with legs $3$ and $4$. Here's what I mean $$3\sin t + 4\cos t = \sqrt{3^2+4^2}\frac{3\sin t+4\cos t}{\sqrt{3^2+4^2}}=5\bigg(\frac{3}{5}\sin t + \frac{4}{5}\cos t\bigg)$$ Now I think that you can see that the sum of the squares of the coefficients in front of the $\sin$ and $\cos$ are now equal to 1. $$\bigg(\frac{3}{5}\bigg)^2+\bigg(\frac{4}{5}\bigg)^2=\frac{9+16}{25}=1$$ So now the problem is in a form were we can say that $$\cos\alpha = \frac{3}{5}$$ and $$\sin\alpha = \frac{4}{5}$$ I think that that should be enough of an explanation for you to figure out how to handle amplitudes and phase shifts in similar problems. I think that you can take it from here.