Question

What does `sin(arccos(5))-3sec(arc sin(8))` equal?

Answer 1

Nothing meaningful.

Explanation:

The arguments for "arccos" and "arcsin" must be within the range `[-1,+1]`. The values given are not valid.

The argument of "arccos" must be a value which could be generated by the `cos` function and the "cos" function only generates values in the range `{-1,+1]`.

Similarly for "arcsin".

Answer 2

Using definitions of Complex `cos`, `sin`, etc.:

`sin(arccos(5)) - 3sec(arcsin(8))=(2sqrt(6)+sqrt(7)/7)i`

Explanation:

While Alan's answer is correct for `sin` and `cos` considered as Real valued functions of Real arguments, it is possible to define them as Complex valued functions of Complex arguments:

`cos z = (e^(iz)+e^(-iz))/2`

`sin z = (e^(iz)-e^(-iz))/(2i)`

Note in passing that `sin^2 z + cos^2 z = 1` for any Complex number `z`.

With these definitions, it is possible to define `arccos` and `arcsin` for values other than `[-1, 1]`.

`color(white)()`
If `cos z_1 = 5` then `sin^2 z_1 = 1 - cos^2 z_1 = 1 - 25 = -24`

Hence `sin(arccos(5)) = sqrt(-24) = 2sqrt(6)i`

`color(white)()`
If `sin z_2 = 8` then `cos^2 z_2 = 1 - sin^2 z_2 = 1 - 64 = -63`

Hence `cos(arcsin(8)) = sqrt(-63) = 3sqrt(7)i`

`color(white)()`
So:

`sin(arccos(5)) - 3sec(arcsin(8))=2sqrt(6)i - 3/(3 sqrt(7)i)=(2sqrt(6)+sqrt(7)/7)i`