Calculating the dimensions of constants of a displacement function

Question

I'm struggling with the following problem, help would be much appreciated!
The displacement of a moving particle over the x-axis can be described by the following function:
$$x(t) = Ae^{-bt} sin(ct+d)$$
We are using seconds to measure time and meters to measure the displacement.
The constants are:
$$A = 5[?], b = 2[?], c = 4[?], d = 3[?]$$
How can I calculate the dimensions of $A,b,c,d$ and what are the MKS units of $A,b,c,d$?

astromm · Accepted Answer

Remember that the equation needs to be dimensionally consistent (meaning that both sides of the equation have the same units) and that functions that aren't just a variable to some power (e.g. sine, cosine, exponential, etc.) have to have a dimensionless argument.
Dimensional consistency says that $A$ must have the same units as $x$, because the sine and exponential won't carry dimensions. So $A$ has units of meters. Dimensional consistency also says that for quantities to add or subtract, they must have the same units. So $ct$ and $d$ must have the same units.
In order to make the argument of the exponential unitless (if it weren't we could end up with a unit like an exponential meter $e^{meter}$), the units of $b$ and $t$ must cancel. So $b$ has units $1/[t]$=1/s=Hz.
Similarly, the argument of sine should be unitless. However, for trig functions we do keep track of a "unit", namely degrees or radians. The units of the argument of a trig function are naturally radians, so that means $d$ and $ct$ have units of radians. We know the units of $t$ are seconds, so this means $c$ has units rad/s.
Hope this helps!

Gert · Answer

A $sin$ function always returns a dimensionless number.
By consequence $A$ must be expressed in  $mathrm{m}$, because $x(t)$ is.
The argument of a $sin$ function is also dimensionless (some say $mathrm{radians}$ but that is dimensionless). So:
$$ct+d$$
is dimensionless, which means that $d$ is dimensionless and:
$$ct$$
is dimensionless too. With $t$ in $mathrm{s}$, then $c$ must be in $mathrm{s^{-1}}$.
The exponent of an exponential function (like $e^{-bt}$) is dimensionless, so $b$ must have dimensions $mathrm{s^{-1}}$.

Calculating the dimensions of constants of a displacement function

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