Simplify sin(arctan(u))

Problem

sin(arctan(u))

Solution

1. Define a right triangle where θ=arctan(u) which implies tan(θ)=u

2. Identify the sides of the triangle based on tan(θ)=opposite/adjacent We can let the opposite side be u and the adjacent side be 1

3. Calculate the hypotenuse using the Pythagorean theorem: a2+b2=c2

c=√(,1+u2)

c=√(,1+u2)

4. Apply the definition of sine, which is sin(θ)=opposite/hypotenuse

sin(θ)=u/√(,1+u2)

Final Answer

sin(arctan(u))=u/√(,1+u2)

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