Find the Exact Value arcsec(-9)

Problem

arcsec(−9)

Solution

1. Identify the definition of the inverse secant function. The expression y=arcsec(−9) is equivalent to finding an angle y such that sec(y)=−9 where y is in the interval [0,π] and y≠π/2

2. Relate the secant function to the cosine function using the reciprocal identity sec(y)=1/cos(y)

3. Substitute the given value into the identity to find the corresponding cosine value:

cos(y)=−1/9

4. Express the angle y using the inverse cosine function:

y=arccos(−1/9)

5. Determine if the value can be simplified further. Since −1/9 is not a coordinate of a standard angle on the unit circle, the exact value is expressed using the inverse function notation.

Final Answer

arcsec(−9)=arccos(−1/9)

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