Evaluating a Trigonometric Expression

Date: 07/31/99 at 10:50:26
From: Alireza Rabi
Subject: Evaluation of a Trigonometric Expression

Can you help me evaluate tan(arcsin(3/5))? I think first we have to 
find the angle for which the sine is 3/5, and then take the tangent of 
it. We have to solve this problem with the sin^2(x)+cos^2(x) = 1. 

Date: 07/31/99 at 20:53:16
From: Doctor Peterson
Subject: Re: Evaluation of a Trigonometric Expression

You have made a good start. Everything you have said is indeed part of 
the solution.

Here is how I do this kind of problem, so that I do not have to 
explicitly use sin^2(x)+cos^2(x) = 1 or do any algebra, but just sort 
of "see" the answer.

Draw a triangle (it does not have to have the right dimensions):

         +
        /|
       / |
      /  |
     /A  |
    +----+

Now if we know the sine of A is 3/5, we can label the triangle:

         +
        /|
       / |
     5/  |3
     /A  |
    +----+

Now you can use the Pythagorean Theorem (which is equivalent to the 
identity you stated) to find the missing side, and from that you can 
find the tangent.

Let me know if you need more help.

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