Find the Exact Value arccsc(-1)

Problem

arccsc(−1)

Solution

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

2. Rewrite the expression in terms of the sine function. Since csc(y)=1/sin(y) the equation csc(y)=−1 is equivalent to 1/sin(y)=−1

3. Solve for sin(y) by taking the reciprocal of both sides.

sin(y)=−1

4. Determine the angle y within the restricted range [−π/2,π/2] that satisfies the equation. The sine of an angle is −1 at y=−π/2

Final Answer

arccsc(−1)=−π/2

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