Expert: Paul Klarreich - 1/29/2007

Question
I know this is supposed to be easy, but it is hard to understand our professor sometimes with his thick French Accent.

The question is, find the exact value of the expression sec(arctan 2).

Can you describe an easy step by step way of doing these types of problems...  

Answer
Questioner:   Aaron
Category:  Calculus
 
Subject:  Inverse Trigonometric Functions
Question:  I know this is supposed to be easy, but it is hard to understand our professor sometimes with his thick French Accent.

The question is, find the exact value of the expression sec(arctan 2).

Can you describe an easy step by step way of doing these types of problems...
................................................
Hi, Aaron,

Ze scheme is seemple, mon ami.  You start by writing ze  arctan 2 as theta. because after all, eet ees an angle, no?

theta = arctan 2.

Then  tan(theta) = 2, so you proceed:

A. Construct a triangle ABC, where:

  Angle C is a right angle.
  Angle A = theta.
  Side  BC = 2
  Side  AC = 1,

so now tan(theta) = 2/1 = 2, as required.  

[This is the general scheme to find any trig function of theta, given any other trig function of theta.]

B. Use the Pythagorean Theorem to find that AB = sqrt(5).
                    AB    sqrt(5)
C. Now  sec(theta) = -- =  ------- = sqrt(5)
                    AC       1

There are other ways to do it, but all of them start by writing the key fact:

theta = arctan 2,  and  therefore  tan(theta) = 2.

You could also use an identity:  sec^2 t = 1 + tan^2 t

Then   sec^2 t = 1 + (2)^2 = 5,  so sec t = sqrt(5)