Find the Exact Value arccsc(- square root of 2)

Problem

arccsc(−√(,2))

Solution

1. Identify the definition of the inverse cosecant function. Let y=arccsc(−√(,2)) This is equivalent to finding y such that csc(y)=−√(,2) within the range [−π/2,0)∪(0,π/2]

2. Convert the cosecant equation into a sine equation using the reciprocal identity csc(y)=1/sin(y)

sin(y)=1/(−√(,2))

3. Rationalize the denominator of the fraction.

sin(y)=−√(,2)/2

4. Determine the angle y in the interval [−π/2,π/2] that satisfies the sine equation. Since sin(π/4)=√(,2)/2 and the sine function is odd, sin(−π/4)=−√(,2)/2

Final Answer

arccsc(−√(,2))=−π/4

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