What is the connection between $RX$ gates and $X$ gates (similar for $Y$ and $Z$)?

Question

I am new to quantum gates but do not understand the connection between the $RX$ and $X$ gates. I know that
$$R X(theta)=exp left(-i frac{theta}{2} Xright)=left(begin{array}{cc}
cos frac{theta}{2} & -i sin frac{theta}{2} 
-i sin frac{theta}{2} & cos frac{theta}{2}
end{array}right)$$
Meanwhile the $X$ gate is given by
$$X = left(begin{array}{cc}
0 & 1 
1 & 0
end{array}right)$$
Is there a value of $theta$ such that the two are the same? I see that choosing $theta = pi/2$ gives the result upto an overall factor of $-i$. Is that it or is there a deeper connection between the two gates? Is there a similar connection between the $Y$ and $RY$ gates and the $Z$ and $RZ$ gates such that the rotated gates are more general than the $X, Y$ and $Z$ gates?

C. Kang · Accepted Answer

You're almost correct - choosing $ theta = pi$ does yield $$ begin{bmatrix} 0 & -i  -i & 0 end{bmatrix} $$
Because this differs from the $X$ gate by a constant factor global phase ($ -i$), the gates are equivalent. (See here to learn more about the global phase).
This connection holds similarly for $ RY$ and $Y$, and $RZ$ and $Z$. A way to visualize this is the Bloch sphere: in essence, these gates are rotations about the $X, Y, Z$ axes (respectively):

So essentially our Pauli primitives are $pi$ rotations over the respective axis.

What is the connection between $RX$ gates and $X$ gates (similar for $Y$ and $Z$)?

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