Write the composed trigonometric function sin(arctan x) in terms of x. Explain your steps and/or show your work. Remember to rationalize the denominator if necessary.

Answer 1

We start with the trigonometric identity:

sin(arctan x) = sin(θ) where θ = arctan x

Let's draw a right triangle with angle θ = arctan x:

Let's call the opposite side y, the adjacent side 1, and the hypotenuse r:

Then, we have:

tan θ = y/1
tan θ = y

Using Pythagoras theorem, we have:
r = sqrt(1^2 + y^2)
r = sqrt(1 + y^2)

Then, we have:
sin(θ) = y/r
sin(arctan x) = y/sqrt(1 + y^2)

Substitute y = tan θ = x:

sin(arctan x) = x/sqrt(1 + x^2)

Therefore, the composed trigonometric function sin(arctan x) = x/sqrt(1 + x^2).

We rationalize the denominator by multiplying both numerator and denominator by sqrt(1 - x^2):

sin(arctan x) = x * sqrt(1 - x^2) / (1 + x^2)