Simplify csc(arctan(( square root of 3)/3))

Problem

csc(arctan(√(,3)/3))

Solution

1. Identify the inner value √(,3)/3 as a known trigonometric ratio.

2. Rationalize or recognize that √(,3)/3 is equivalent to 1/√(,3)

3. Evaluate the inner function θ=arctan(1/√(,3)) which represents the angle in the interval (−π/2,π/2) whose tangent is 1/√(,3)

4. Determine that θ=30 or π/6 radians.

5. Substitute the angle back into the outer function to get csc(30)

6. Calculate the cosecant value using the identity csc(θ)=1/sin(θ)

7. Evaluate sin(30)=1/2 so csc(30)=1/(1/2)=2

Final Answer

csc(arctan(√(,3)/3))=2

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