Simplify arccsc(- square root of 2)

Problem

arccsc(−√(,2))

Solution

1. Identify the definition of the inverse cosecant function. The expression y=arccsc(x) is equivalent to csc(y)=x where y is in the range [−π/2,0)∪(0,π/2]

2. Rewrite the equation using the sine function. Since csc(y)=1/sin(y) the equation csc(y)=−√(,2) is equivalent to:

sin(y)=−1/√(,2)

3. Rationalize the value on the right side of the equation to make it more recognizable.

sin(y)=−√(,2)/2

4. Determine the angle y within the restricted range [−π/2,π/2] that satisfies the sine equation. We know that sin(π/4)=√(,2)/2

5. Apply the property of odd functions, sin(−y)=−sin(y) to find the negative value.

sin(−π/4)=−√(,2)/2

Final Answer

arccsc(−√(,2))=−π/4

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