Precalculus teacher vs WolframAlpha student

The domain of a function is a big part of precalculus. Cos^-1(pi) is actually an imaginary number! Learn more fundamentals of math at Brilliant! Use https://brilliant.org/blackpenredpen/ to get 20% off!

The restricted domain to have an inverse for cos(x) is [0, pi] and its range is [-1,1]. Thus we know the domain of the inverse of cos(x) is [-1, 1] and the range is [0,pi]. With that said, cos^-1(pi) shouldn't have any answer likewise for cos(cos^-1(pi)). But one of my students used WolframAlpha and it gave cos(cos^-1(π))=π!!! Oh well, let's use the complex definition of cos(z) to find an expression for the complex cos^-1(w) and see how we can get the π! Cancel with caution (from my precalculus & calculus tutorial channel):    • Cancel with caution!  

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0:00 cos(cos^-1(pi))=?
1:20 Using Euler's formula to get complex cos(z)
2:47 Getting an expression for complex cos^-1(w)
6:02 A complex value for cos^-1(π)
6:35 Getting cos(cos^-1(π))=π
10:32 check out Brilliant

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