Simplify arcsec(2)

Problem

arcsec(2)

Solution

1. Identify the definition of the inverse secant function. The expression y=arcsec(2) is equivalent to finding an angle y such that sec(y)=2 within the restricted range [0,π/2)∪(π/2,π]

2. Convert the secant equation into a cosine equation using the reciprocal identity sec(y)=1/cos(y)

cos(y)=1/2

3. Determine the angle in the first quadrant whose cosine value is 1/2 From the unit circle or standard trigonometric values, we know that cos(π/3)=1/2

4. Verify that the angle π/3 falls within the range of the principal value for the inverse secant function. Since 0≤π/3<π/2 it is the correct value.

Final Answer

arcsec(2)=π/3

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