How to Solve Inverse Trig Functions Without a Calculator?

[gech]

Supposing I need to solve a problem like: sec(arctan 2) or cos(2arcsin(5/13)), is there a method I could use that would not require a calculator? What I mean is that for an example like tan(arccos .5), the answer is "simple" because I know the arc cosine of .5 is pi/3 and then the tan of pi/3 is the squareroot of 3. But in a problem like the two above, how would I go about doing this where the numbers are more complicated and I want to do it by hand? Or is it impossible without the use of a calculator?

 

[Tide]

Here's one way: In the case of sec(atan(2)) draw a right triangle and label one of the angles, say A. Label the side opposite A with 2 and the side adjacent to it with 1. Clearly, A is an angle whose tangent is 2. What's the secant of this angle? Use Pythagoras to find the length of the hypotenuse and divide it by 1 (adjacent side) to find sec(A) = sec(atan(2)).

You can do similar things for your other example by applying well known trig identities.

 

[gech]

I see what you're getting at. Thank you.