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### Trig Inverses: sin(arctan x)

```Date: 11/04/2002 at 15:06:32
From: Erin
Subject: Trig Inverses

We got a question in my trig class today, and I don't know where to
start. We discussed the inverses of trig functions, but when the
teacher threw an x in there, I didn't understand. Here is the problem
she gave us.

sin(arctan x)

```

```
Date: 11/05/2002 at 14:05:41
From: Doctor Ian
Subject: Re: Trig Inverses

Hi Erin,

When you have an expression like

sin(x)

it means that if you supply the value of an angle, the function will
return a corresponding value:

sin(45 degrees) = sin(pi/4 radians) = sqrt(2)/2

So if you have

sin( f(x) )

it implifes that f(x) is yet another kind of function, that takes some
value of x and returns an angle, which will then be the argument to
the sine function.

Does that make sense so far?

Well, arctan is just such a function. It takes a value, such as 1, and
returns the angle that _would_ give you that value, if used as an
input to the tangent function.  So

1)  tan(45 deg) = 1

2)  arctan(1) = 45 deg

3)  tan( arctan(1) ) = tan( 45 deg ) = 1

4)  arctan( tan( 45 deg) ) = arctan(1) = 45 deg

The other common name for 'arctan' is 'inverse tangent', and these
examples show why: the tan() and arctan() functions cancel each other
out (although you have to be careful to keep track of what quadrant

The expression

sin(arctan x)

means

give me some value; arctan will tell you for which angle
tan(x) = that value; and sin(...) will tell you the sine
of the angle.

Let's work through it for the following triangle,

|\
a | \ c
|__\
b  A

using x = (a/b) for our argument:

sin( arctan (a/b) )

= sin( A )                 because tan(A) = a/b,
so arctan(a/b) = A

= a/c                      because sin(A) = a/c

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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