Simplify sec(arctan(3x))

Problem

sec(arctan(3*x))

Solution

1. Identify the relationship between the trigonometric function and its inverse. Let θ=arctan(3*x) which implies tan(θ)=3*x

2. Represent the relationship using a right triangle where the angle is θ Since tan(θ)=opposite/adjacent we can set the opposite side to 3*x and the adjacent side to 1

3. Apply the Pythagorean theorem to find the hypotenuse h

h=√(,(3*x)2+1)

4. Simplify the expression for the hypotenuse.

h=√(,9*x2+1)

5. Evaluate the secant of the angle θ Since sec(θ)=hypotenuse/adjacent substitute the values from the triangle.

sec(θ)=√(,9*x2+1)/1

Final Answer

sec(arctan(3*x))=√(,9*x2+1)

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