As i read MD5 have many complex formulas, example:
$k_i := \lfloor \space| \sin(i + 1) \times 2^{32}| \space\rfloor$
My question is why use sin(), how about cos() or tan() or something like that. Does it make any sense when use sin() ? Or just for more complex of formula lead to more secure of algorithm ?
In MD5, the use of formulas with floating point arithmetic is limited to building 64 nothing-up-my-sleeves 32-bit constants. Most implementations use a precomputed table, for startup efficiency and portability.
Changing $\sin$ to $\cos$ would change the constants, but as far as we know would make the algorithm neither weaker nor stronger. That's the nature of nothing-up-my-sleeves constants.
There would be a problem with using $\tan$, because $\tan(11)\approx-225.95$, thus the input of floor would be 40-bit. We would need a floor variant with at-least 40-bit result, keep the low-order 32-bit part of the result, and ensure that $\tan$ is evaluated to at least 40-bit precision in the mantissa for portable values from an implementation to another¹.
MD5 has other arbitrary-looking formulas, operating on 32-bit unsigned integers assimilated to 32-bit vectors. These are optimized for their cryptographic properties, and changing them haphazardly stands high risk of weakening the algorithm (further than it is: its security goal of collision-resistance is not met).
¹ Incidentally: computing the MD5 constants using the built-in floating point arithmetic in the Arduino will fail to yield the correct table due to insufficient (non-conforming) precision of the double type there, and would cause the low-order 8 bits to be severely biased, with possible negative cryptographic consequences. The same effect causes many Arduino code using a GPS to be significantly less accurate than it should be. But I digress.
