Inverse Trig Functions and Special Angles

Date: 01/25/2006 at 18:29:16
From: Alan
Subject: inverse trig functions

Find the exact value of the expression arctan(sqrt(3)).

In the solutions manual for this problem it says tan(pi/3) = sqrt(3)
but they don't say how they figured that out.  I need to know how to
figure that out without a calculator or graph because neither one is
allowed on exams.

I know that y = arctan(x) means that tan(y) = x and therefore 
y = arctan(sqrt(3)) means tan(y)=sqrt(3).  How do I know what y is?

Date: 01/26/2006 at 08:22:39
From: Doctor Fenton
Subject: Re: inverse trig functions

Hi Alan,

Thanks for writing to Dr. Math.  You are generally expected to know
the values of the trigonometric functions at "special angles", which
usually refers to those angles arising from the equilateral triangle
(and the bisected equilateral triangle, which gives you a 30º-60º-90º
triangle) and the isosceles right triangle.  That means that you 
should know the exact value of sin, cos, and tan of 0, pi/6 (30º),
pi/4 (45º), pi/3 (60º), and pi/2 (90º) (as well as any angle whose
reference angle is one of these).  There is a chart at:

  Trigonometry Formulas
    http://mathforum.org/dr.math/faq/formulas/faq.trig.html 

A mnemonic I like to use is what I call the "half square root" table:

Write a blank chart

      x         sin(x)             cos(x)
     --------------------------------------
      0         sqrt( )/2         sqrt( )/2  
      pi/6      sqrt( )/2         sqrt( )/2  
      pi/4      sqrt( )/2         sqrt( )/2   
      pi/3      sqrt( )/2         sqrt( )/2  
      pi/2      sqrt( )/2         sqrt( )/2     .


Now, just fill in the blanks with the numbers 0 through 4, going
downward in the sin(x) column and upward in the cos(x) column.  If you
are familiar with the sine and cosine curves, you can remember that in
the first quadrant the sine starts at 0 and rises to 1, while the
cosine starts at 1 and falls to 0, which helps remember which way to
put in the 0-4 values in this chart.

      x         sin(x)             cos(x)
     --------------------------------------
      0         sqrt(0)/2         sqrt(4)/2  
      pi/6      sqrt(1)/2         sqrt(3)/2  
      pi/4      sqrt(2)/2         sqrt(2)/2   
      pi/3      sqrt(3)/2         sqrt(1)/2  
      pi/2      sqrt(4)/2         sqrt(0)/2    .


When you simplify, you get the usual chart for sin(x) and cos(x):

      x         sin(x)             cos(x)
     --------------------------------------
      0           0                   1
      pi/6       1/2              sqrt(3)/2 
      pi/4     sqrt(2)/2          sqrt(2)/2 
      pi/3     sqrt(3)/2             1/2 
      pi/2        1                   0  .

With this chart, you can find tan(x) and the value of any of the 
reciprocal functions cot(x), sec(x), and csc(x) by using the identity 
tan(x) = sin(x)/cos(x) and the necessary reciprocal identity (such as 
sec(x) = 1/cos(x) ).

If you have any questions, please write back and I will try to 
explain further.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/