sec = 1/cos
so what angle has cosine = -1/2
that angle is 180 + or - 60 degrees
which is pi +/- pi/3
2 pi/3 or 4 pi/3
arcsec(-2) is the same angle as arccos(-1/2)
The angle 2pi/3 is correct, so is 4pi/3.
Depending on the domain required, you may have to put in both values (for 0-2pi).
If that is not the problem, then you have to read carefully the instructions how to answer the question, how to type in the %pi; symbol, or the number of significant figures, etc.
since cos pi/3 = 1/2,
sec pi/3 = 2
But, arcsec has principal values from 0 to pi, so arcsec(-2) = pi - pi/3 = 2pi/3.
Note: Some authors define the range of arcsecant to be ( 0 ≤ x < π/2 or π ≤ x < 3π/2 ), because the tangent function is nonnegative on this domain. This makes some computations more consistent.
So, you'd better check to see how your course defines the principal values. Apparently you were expected to provide 4pi/3 as the arcsec(-2)
Perhaps you were supposed to give both the quadrant 2 and quadrant 2 answers
Oh, thank you so much Steve. I just found a very old sheet from class with the restrictions and you are right-- my teacher gave the unconventional one. I studied from online sources, which all gave the other range, so that's why I got it wrong. Thanks!