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

To find the composed trigonometric function sin(arctan(x)), we will first use the identity involving inverse trigonometric functions:

sin(arctan(x)) = sin(θ) = opposite/hypotenuse = x/√(1+x^2)

To justify this, consider a right triangle with angle θ such that tan(θ) = x. Then, the opposite side of the triangle is x and the hypotenuse is 1, so the sine of θ is x/√(1+x^2).

Therefore, sin(arctan(x)) = x/√(1+x^2) is the simplified expression for the composed trigonometric function.