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What is Arctan?

Date: 8/28/96 at 11:15:0
From: Anonymous
Subject: What is Arctan?

I don't understand a question. it says:

Let y = f(x) = sin (arctan x).  

Is sin the same as arctan X?  Is it something different?  Do I 
multiply the two together?

Thank You!

Date: 10/17/96 at 10:58:10
From: Doctor Roger
Subject: Re: What is Arctan?

The sine and the arc-tangent are different functions, and both require
arguments to make sense (argument means the input to the function). In 
your equation the argument of the Arctan is (x), while the argument of 
the sin is (Arctan x). They are *not* multiplied together. The 
notation means to take the arctangent of x first, then find the sine 
of the result.

In order to simplify the equation, first consider arctan x. It 
represents that angle whose tangent is x. For example, if theta = 
arctan(x), then x = tan(theta). Let's make a diagram of a triangle 
having an angle theta, such that the tangent of theta is x:

                   .  |
                 .    |
             H .      | x
             .        |
           .          |
         . theta      |

Notice that the Pythagorean Theorem tells us that the hypotenuse H is
given by H = sqrt (1+x^2). Thus we have:
                         f(x) = sin (arctan (x))
                              = sin (theta)
                              = x / H
                              = x / sqrt(1+x^2)

You might object that the "size" of my triangle was irrelevant - and
that is right! I could just as easily have picked a triangle with 
sides of 2x and 2, or any other side lengths that are in the ratio of 
x:1. (Remember - the tangent of theta has to be x.) The hypotenuse 
would have scaled accordingly, and we would have come up with the same 
expression for sin(arctan(x)). (Try it!)

-Doctor Roger,  The Math Forum
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Associated Topics:
High School Trigonometry

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