Some modern texts on calculus or real analysis define the functions in the title (with appropriate domain inside $\mathbb{R}$) using Riemann integration.
However, it seems that these functions may have been used by Mathematicians before Bernhard Riemann, with different definitions. The functions $\sin x$, $\cos x$ appeared in geometry as well as trigonomety, which are quite old subjects, than analysis.
Since in the classes of calculus or analysis, almost everyone start these functions with Definition: The function ... defined by ... is called ... Also, very less books describe such functions with some historical comments. It seems appropriate to post a question here, and I wonder to know a little bit of early definitions of these functions; whether integration is needed to define these functions.
Q. What are other definitions of the functions in title given without using Riemann integration? How these functions were initially defined/considered/understood?
There are a few problems with your question. First of all, the modern notion of functions is quite recent. In the 18th and early 19th century, no mathematician would really have insisted on a strict formal definition of a function like we do today. (see Wikipedia)
Then as a next point, Riemann is kind of a strange point to start things with. While integration gives you a common way of defining the logarithm, integration was known before Riemann, he just gave the definition of a specific type of integral that is commonly taught first today.
Finally, there are many other ways to define those functions which are also used in modern courses but have been known for quite a while:
The most common definition of the exponential function is via its power series $$exp(x) = \sum_{k=0}^\infty \frac{x^k}{k!}$$ It is not hard to see that this converges. Since this results in a monotone function, it has an inverse, this inverse is then called $\log$.
For $x \in \mathbb{Q}$, $a>0$ the power $a^x$ is well defined by algebra. This has an unique continuous extension to $x\in\mathbb{R}$. Another way of course is just using $a^x := \exp(\log(a) x)$, which of course is consistent with the other definition. From this point you can of course also show that $\exp(x) = e^x$.
Finally $\sin$ and $\cos$ I have mostly seen defined via their power series. However the age old definition by using the lengths of the sides of a right triangle with a certain angle is fine as well.