Take this equation $$v^2 = 2 a \sin(2\pi b t)$$ where $v$ is a velocity and $t$ is a time. Find the dimensions of $a$ and $b$.
Finding the dimensions of $a$ is easy as the $\sin$ has no dimension so it will have the same dimensions as $v^2$.
But the problem is finding the dimensions of $b$ because the $\sin$ takes an angle as a parameter, and angles have no dimension we can simply say that $b$ should be $1/[\mathrm{T}]$ which will make the entire argument dimensionless. But isn't that just wrong?
Because there are some physical laws that have $\sin(t)$ for example where $t$ is a time, so it isn't a condition that the angle's value should be dimensionless.
This is really confusing.
Answer
The arguments of trigonometric functions in physical equation have always to be dimensionless because the argument is a pure number. Therefore you are correct that the dimension of b has to be [1/T].
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