{"id":6009,"date":"2011-05-03T09:29:35","date_gmt":"2011-05-03T01:29:35","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=6009"},"modified":"2016-09-19T11:13:31","modified_gmt":"2016-09-19T03:13:31","slug":"which-is-the-correct-graph-of-arccot-x","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/which-is-the-correct-graph-of-arccot-x-6009","title":{"rendered":"Which is the correct graph of arccot x<\/em>?"},"content":{"rendered":"

A reader challenged me on the graph I had for y<\/em> = arccot x<\/em>,<\/span> on the page Inverse Trigonometric Functions<\/a>. <\/p>\n

He wrote (in a rather unfriendly tone):<\/p>\n

\n

Compare:<\/p>\n

https:\/\/en.wikipedia.org\/wiki\/Inverse_trigonometric_functions<\/a><\/p>\n

https:\/\/www.intmath.com\/analytic-trigonometry\/7-inverse-trigo-functions.php<\/a><\/p>\n

You incorrectly state the range of the arccotangent function as -pi\/2 to pi\/2.  It is not.  The correct range of arccotangent is 0 to pi. <\/p>\n<\/blockquote>\n

After some deliberation, I have now included both interpretations on my page, because both are found in various sources.<\/p>\n

First, some background.<\/p>\n

Obtaining the graph of y<\/em> = arccot(x<\/em>) <\/h2>\n

The graph of y<\/em> = arccot x<\/em><\/span> can be obtained from a consideration of the graph of y<\/em> = cot x<\/em>.<\/span> <\/p>\n

But depending on your starting region, you'll get a different graph for y<\/em> = arccot x<\/em>.<\/span><\/p>\n

Interpretation 1<\/h2>\n

The graph of y<\/em> = cot x<\/em><\/span> is as follows:<\/p>\n

\"cot<\/p>\n

We choose the portion from x<\/em> = 0<\/span> to x<\/em> = π<\/span> (as highlighted above), and reflect it in the line y = x<\/em><\/span> like this.<\/p>\n

\"cot<\/p>\n

Since reflection in the line y = x<\/em><\/span> gives us the inverse of a function, we have obtained the graph of y<\/em> = arccot x<\/em><\/span>, as follows:<\/p>\n

\"arccot<\/p>\n

From the graph we can see the domain<\/strong> (the possible x<\/em>-values) of y<\/em> = arccot x<\/em><\/span> is: <\/p>\n

All values of x<\/em><\/p>\n

And the range<\/strong> (resulting y-<\/em>values) of arccot x <\/em>is: <\/p>\n

0 < arccot x<\/em> < π <\/span><\/p>\n

If we evaluate our function for some negative value of x<\/em>, say x<\/em> = −2, then we get a positive answer, as expected from the graph:<\/p>\n

arccot(−2) = 2.678...<\/span> <\/p>\n

<\/ins><\/div>\n

Alternate View - Interpretation 2 <\/h2>\n

Some math textbooks (and some respected math software, e.g. Mathematica<\/em>) regard the following as the region of y<\/em> = cot x<\/em><\/span> that should be used (that is, −π\/2<\/span> to π\/2<\/span>): <\/p>\n

\"cot<\/p>\n

This would give the following discontinuous graph when reflected in the line y = x<\/em><\/span>:<\/p>\n

\"arccot<\/p>\n

So the domain<\/strong> of arccot x<\/em><\/span> would be (as for Interpretation 1): <\/p>\n

All values of x<\/em> <\/p>\n

Using this interpretation, the range<\/strong> of arccot x<\/em> would be:<\/p>\n

−π\/2 < arccot x<\/em> ≤ π\/2 <\/span>(arccot x<\/em> ≠ 0<\/span>) <\/p>\n

If this is the correct graph, we expect a negative answer when we evaluate the function at x<\/em> = −2. It is actually:<\/p>\n

arccot(−2) = −0.46365...<\/span><\/p>\n

Which is correct?<\/h2>\n

According to a response to a reader's question<\/a> on this same issue, Dr. Math goes for the first<\/strong> interpretation: <\/p>\n

\n

In order to invert a trig function, we first restrict it to a domain on which it takes all its possible values, once each; then we invert the restricted function, whose range is then that restricted domain.<\/p>\n

Look at a graph of the cotangent function, and you will see that although between -pi\/2 and pi\/2 it takes all its possible values, and takes each value only once, there is one problem with this choice: it is not continuous (or even defined) on this entire domain, but is undefined at 0. <\/p>\n

The domain would then have to be<\/p>\n

-pi\/2 < x < 0 or 0 < x <= pi\/2 <\/p>\n

To avoid this, we instead choose the domain<\/p>\n

0 < x < pi<\/p>\n

which is cleaner to work with, making a continuous function defined over the entire domain.<\/p>\n<\/blockquote>\n

Math software doesn't agree, either<\/h2>\n

Let's now see the inconsistent way (respected) math software deals with this function.<\/p>\n

Mathcad's Interpretation <\/h2>\n

When I graphed the function acot(x<\/em>)<\/span> in Mathcad, this is the result (they are using the first<\/strong> interpretation): <\/p>\n

\"arccot<\/p>\n

Mathcad gives arccot(−2) = 2.678...,<\/span> which is consistent with their graph. Mathcad uses "acot(x<\/em>)" notation. <\/p>\n

Mathematica's Interpretation <\/h2>\n

However, according to Mathematica<\/a> (using Wolfram|Alpha) , this is the graph of y<\/em> = arccot(x<\/em>)<\/span>:<\/p>\n

\"arccot<\/p>\n

So Mathematica is using the second<\/strong> interpretation of the function. (They use the somewhat poor notation y<\/em> = cot−1<\/sup>x<\/em>.<\/span>) <\/p>\n

Taking a typical value, Mathematica (Wolfram|Alpha) gives us<\/p>\n

arccot(−2) = −0.46365...<\/span> <\/p>\n

How can this be? The first interpretation gives us a positive value for arccot(−2), while the second interpretation gives us a negative value.<\/p>\n

Matlab's Interpretation <\/h2>\n

Matlab also gives us a discontinuous graph based on the second<\/strong> interpretation. <\/p>\n

\"arccot <\/p>\n

Matlab also uses "acot(x<\/em>)"<\/span> notation. <\/p>\n

Maple\/Scientific Notebook's Interpretation<\/h2>\n

My version of Scientific Notebook has both Maple and MuPAD engines. The results using both of these are curious.<\/p>\n

Using the Maple engine, we get an answer using the first<\/strong> interpretation. <\/p>\n

\"arccot <\/p>\n

But when we switch to use MuPAD in Scientific Notebook, we get this result, which uses the second<\/strong> interpretation!<\/p>\n

\"arccot <\/p>\n

So even the top math software makers can't agree.<\/p>\n

Conclusion<\/h2>\n

Here's another case where our math definitions are not as tight as we are often led to believe.<\/p>\n

Also, the notation is inconsistent. Different text books and different software use the following to mean the same thing:<\/p>\n