Answer: Let's start by using the identity:
tan(arctan(x)) = x
to simplify the expression inside the sine function. So, we have:
arctan(u) / sqrt(3) = tan(arctan(u) / sqrt(3))
Now, using the trigonometric identity:
tan(x/y) = sin(x) / (cos(y) + sin(y))
with x = arctan(u) and y = sqrt(3), we get:
tan(arctan(u) / sqrt(3)) = sin(arctan(u)) / (cos(sqrt(3)) + sin(sqrt(3)))
Simplifying further, we know that:
sin(arctan(u)) = u / sqrt(1 + u^2)
and
cos(sqrt(3)) + sin(sqrt(3)) = 2cos(sqrt(3) - pi/4)
So, the expression becomes:
sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2cos(sqrt(3) - pi/4)]
Simplifying the denominator, we have:
sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / (2(cos(sqrt(3))cos(pi/4) + sin(sqrt(3))sin(pi/4)))
Using the values for cosine and sine of pi/4, we get:
cos(pi/4) = sin(pi/4) = 1/sqrt(2)
So, we have:
sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2(sqrt(3)/2 + 1/2)]
Simplifying further:
sin(arctan(u) / sqrt(3)) = u / (sqrt(1 + u^2) * (sqrt(3) + 1))
Therefore, the algebraic expression for sin(arctan(u) / sqrt(3)) is:
u / (sqrt(1 + u^2) * (sqrt(3) + 1))
Explanation: