How do you calculate #tan^-1 (12.4304)#?

One can compute #\tan^{-1}(12.4304)# using calculator which gives

#\tan^{-1}(12.4304)=85.4^\circ#

#color(blue)("The teaching bit")#

Within the context of this question if you take tangent of the angle #theta# you obtain the value #12.4304#

Writing: #tan^(-1)(12.4304)# means that you are asking: What is the angle whose tangent is 12.4304

Another way of writing #color(purple)(tan^(-1)(12.4304))" "# is #" "color(purple)(arctan(12/4304))#

They both mean the same thing. I much prefer the second one as there is no confusion as to what it means when someone first comes across the format #tan^(-1)(12.4304) #

They, in error, could think this means #1/tan(12.4304)#.

#color(magenta)("IT DEFINITELY DOES NOT MEAN THAT!")#
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#color(white)()#

#color(blue)("Answering the question")#

#color(brown)("What is the value of "arctan(12.4304))#

In this case the tangent is the ratio #(b/a) -> 12.4304/1= 12.4304#

The amount of up or down for the amount of 1 along.

This should sound familiar!

Using the calculator #arctan(12.4304)~~85.4005781.....#

Approximately #85.40# degrees rounded to 2 decimal places.