Simplify sin(arctan(3x))

Problem

sin(arctan(3*x))

Solution

1. Identify the inner function as an angle θ=arctan(3*x) which implies tan(θ)=3*x

2. Represent the relationship using a right triangle where the side opposite to θ is 3*x and the side adjacent to θ is 1

3. Calculate the hypotenuse h using the Pythagorean theorem a2+b2=h2

h=√(,(3*x)2+1)

h=√(,9*x2+1)

4. Apply the definition of the sine function, which is the ratio of the opposite side to the hypotenuse.

sin(θ)=(3*x)/√(,9*x2+1)

Final Answer

sin(arctan(3*x))=(3*x)/√(,9*x2+1)

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