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### Inverses of Trigonometric Functions

```Date: 08/07/2002 at 22:59:25
From: Jeremy Beasley
Subject: Inverses of Trigonometric Functions

I have a question pertaining to the law of inverses and the
trignometric functions.

In algebra I and II, you learn that functions are defined by applying
the "vertical line" test, meaning that for every value "x" there is
only one corresponding "y" value.

Also, we learned about inverses of functions. To help us understand,
the teacher applied the "horizontal line" test to help us determine
the possibility of a function having an inverse. If you could draw a
horizontal line through a function and the line only intersected once,
then it has a possible inverse. However, if the horizontal line
intersects twice, making it a secant line, then there is no possible
inverse.

This test (if it is valid enough), provokes me to question why the
trig functions cosine and sine have inverses, yet don't pass the
horizontal line test because they are oscillating functions. How is it
possible that these functions can have inverses of arcsin and arccos
respectively?  Are they some type of exception to mathematical law?
And if so, why?
```

```
Date: 08/08/2002 at 09:49:16
From: Doctor Rick
Subject: Re: Inverses of Trigonometric Functions

Hi, Jeremy.

You've asked a good question.

Think about the square root for a moment. The function y = x^2 graphs
as a parabola; the line y = c, where c > 0, cuts the graph in two
places. Thus this function does not have an inverse. But the square
root function is effectively the inverse of the square function. How
do we manage this? We limit the domain of the square function to non-
negative values. In other words, we define a new function:

y = x^2, x >= 0
undefined, x < 0

which graphs as just the right half of the parabola. This function has
an inverse: the square root function. The graph of the square root
function is a parabola turned on its side - but only the top half of
this parabola.

This is a source of confusion for many, as these items in the Dr.
Math archives indicate:

Square Root Function
http://mathforum.org/library/drmath/view/52645.html

Square Root of 100
http://mathforum.org/library/drmath/view/52650.html

We do essentially the same thing with trigonometric functions. They
are not invertible as they stand, so we define new functions that are
only defined on a specified domain:

y = sin(x), -pi/2 <= x <= pi/2     ===> x = arcsin(y)

y = cos(x), 0 <= x <= pi           ===> x = arccos(y)

y = tan(x), -pi/2 < x < pi/2       ===> x = arctan(y)

y = cot(x), 0 < x < pi             ===> x = arccot(y)

y = sec(x),     0 <= x < pi/2      ===> x = arcsec(y)
or -pi <= x < -pi/2

y = csc(x),     0 < x <= pi/2      ===> x = arccsc(y)
or -pi < x <= -pi/2

The limited functions are invertible, and their inverses are arcsin,
arccos, etc.

We may want to refer to ANY angle x whose sine is y, etc. The
notation arcsin(y) is sometimes used in this way. When it is, it does
not represent a function; the term PRINCIPAL VALUE is used for the
single value of arcsin(x), etc. that lies in the range listed above.
A math table I use refers to the principal value of the arcsine of y
as Arcsin(y), using capitalization to indicate the distinction. I
don't know how standard this is. The notation sin^-1(y) should always
indicate the function (returning the principal value).

The choice of range for the principal value can be a source of
confusion. A student recently asked us about arccot, because his
calculator used a range of -pi/2 to pi/2 for the principal value of
the arccotangent, instead of 0 to pi as I listed above, while his
textbook used the range I listed. I found that at least one respected
Web source also used the range -pi/2 to pi/2.

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

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