Are trig functions only defined for unit circles? In my textbooks the trig functions are defined with the help of a unit circle. So does it always have to be a circle with radius $1$ unit? Can't we define trig functions with the help of a circle with some other radius of $2$ or $3$ or $4$ units?
And if suppose it is possible to define trig functions on a circle of radius $2$ units then is this expression still valid: $\sin x:\mathbb{R}\to[-1,1]$?
 A: If we treat trigonometric functions as ratios, then any circle works (in Euclidean space); the ratio of the vertical coordinate of a point on a circle to that circle's radius is equal to the sine of the angle that the ray from the origin to that point makes with the $x$-axis, as measured in the counter-clockwise direction. However, the whole point of the unit circle is to identify sine with $y$ and cosine with $x$. By using the unit circle, and taking the convention that the angle starts at the $x$-axis and goes counter-clockwise, we can use the $x$-coordinate interchangeably with cosine, and $y$ with sine; because the radius is one, taking the ratio between anything and the radius just results in the original value, and thus we can dispense with that part of the definition. So using a circle with radius $2$ would eliminate a simplification that is one of the main motivations for using the unit circle.
However, in a way we can use any circle. Consider the word "unit". It's often used in math to mean "$1$", but if you think of it as literally a "unit", then a unit circle is one in which the basic unit is the radius. If you have a circle with a radius of $1$ meter, then it's a unit circle if you measure every distance in meters. If you have a circle with radius $1$ inch, then it's a unit circle if you measure every distance in inches. So one interpretation of the unit circle is "If you measure the distance from the $x$-axis, and use the radius as your unit, then the number you get is sine". 
A: The fact that you can use any positive radius to define the trigonometric functions, and that you will get the same functions no matter which radius you start with, is a non-trivial fact of Euclidean geometry. It has much to do with similarity of triangles, the fact that $\pi$ is well-defined as the ratio of the circumference of a circle to its diameter (regardless of the radius), and so on. With that in mind, choosing a circle of radius $1$ is a nice normalisation, and yields the particularly nice fact that $(\cos \theta , \sin \theta)$ is the point on the unit circle measuring $\theta$ radians from the positive $X$-axis. 
A: Historically, trigonometric functions were not always defined on a unit circle. Nearly two thousand years ago,
a famous trigonometric table
used a circle of radius $60.$
The function values in that table were mostly greater than $1.$
You could do something similar.
Define a function $\sin_2(\theta)$ so that it is the $y$-coordinate of a point on a circle of radius $2.$
Then the range of $\sin_2$ is $[-2,2].$
Now try to use that function to compute the length of the leg opposite the angle $\theta$ in a right triangle, knowing that the length of the hypotenuse is $15.$ The answer won't be $15 \sin_2(\theta).$
Instead, it will be $\frac{15}{2} \sin_2(\theta).$
You had to divide the hypotenuse by $2$ to compensate for the fact that you used a circle of radius $2$ and thereby doubled all your sine values.
Most of us have decided that this is too much bother and we would rather just use the circle of radius $1.$
I wrote a little more about the history of trigonometry in an answer to a different question.
My conclusion on this question is the same as for that other question:
"in the end it all comes down to (perceived) convenience and usefulness."
A: The unit circle is used for simplicity for the definition of the trigonometric functions but we can obtain the same equivalent definition for a circle with any other radius $R$, indeed by scaling
$$x^2+y^2=R^2 \iff \left(\frac x R\right)^2+\left(\frac y R\right)^2=1 \iff X^2+Y^2=1$$
