Simplify arccsc(-(2 square root of 3)/3)

Problem

arccsc(−(2√(,3))/3)

Solution

1. Rewrite the argument to a more recognizable form by rationalizing the denominator in reverse or simplifying the fraction.

(2√(,3))/3=(2√(,3))/(√(,3)√(,3))=2/√(,3)

2. Relate the cosecant function to the sine function using the identity arccsc(x)=arcsin(1/x)

arccsc(−2/√(,3))=arcsin(−√(,3)/2)

3. Identify the reference angle in the first quadrant where the sine value is √(,3)/2

sin(π/3)=√(,3)/2

4. Apply the range of the inverse sine function, which is [−π/2,π/2] and the property arcsin(−x)=−arcsin(x)

arcsin(−√(,3)/2)=−π/3

Final Answer

arccsc(−(2√(,3))/3)=−π/3

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