How do you evaluate #sec^-1(2)#?

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Answer 1 · 2016-08-13T11:02:52

#sec^{-1}(2) = pi/3#

Let the answer be #x#.

Since #sec^{-1}(2) = x#, #sec(x) = 2#.

As secant is the reciprocal of the cosine function, i.e. #sec(x) = 1/cos(x)#, the following must be true as well.

#1/cos(x) = 2#

Or #cos(x) = 1/2#.

The range of #sec^{-1}(x)# is #[0,pi/2) uu (pi/2,pi]#.
The range of #cos^{-1}(x)# is #[0,pi]#.

Since the ranges are similar, we can find #x# using

#x = cos^{-1}(1/2) = pi/3#