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

2 Answers
Feb 14, 2016

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".

Feb 14, 2016

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#