Evaluating sin(arc-tan x) is a simple process that involves two steps: using a right-angled triangle to label the two sides and the angle in question, which is x, and using the Pythagoras theorem to calculate the remaining side and calculating the function from these values. Writing out the expression in words is the starting point of evaluating it. In this case, it is the sine of Arc-tan x.
Let A represent the value of this angle. So,
A = Arc-tan x
This means that;
tan A = x
Now, what is needed is to find sin A, which can be calculated using a right angled triangle with one of the acute angles as A. Using x = tan A, the sides of the right angled triangle should be labelled; the opposite angle as x and the adjacent angle as 1 because by definition, tan x= Opposite / Adjacent.
The next step is to get the value of the third side, which is the hypotenuse. Using the Pythagoras theorem, the hypotenuse becomes the sqrt(1 + x2). Once the value of the hypotenuse and the two sides are known, any function of angle A can be easily evaluated. In this case, the sine of A is required, which is given by the opposite over hypotenuse. So,
Sine A = x / sqrt(1 + x2)
However, sin A = Sin(Arc-tan x), So,
Sin(Arc-tan x) = x / sqrt(1 + x2), which gives the answer.