Evaluate arcsec( square root of 2)

Problem

arcsec(√(,2))

Solution

1. Identify the definition of the inverse secant function, where y=arcsec(x) means sec(y)=x for y in the range [0,π/2)∪(π/2,π]

2. Rewrite the expression as an equation involving the secant function.

sec(y)=√(,2)

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

cos(y)=1/√(,2)

4. Rationalize the denominator to recognize the standard value from the unit circle.

cos(y)=√(,2)/2

5. Determine the angle y in the interval [0,π] that satisfies the equation.

y=π/4

Final Answer

arcsec(√(,2))=π/4

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