Converting Half-Angle Trig Formulas to Radicals

Update - April 4, 2011: I corrected a typo in Table 2, first column, bottom row.  What was sqrt(6) has been changed to sqrt(5).

In my last blog post, I considered the conversion of  to radicals, a calculation arising from an entry in my Little Red Book of Maple Magic. In another place in this notebook, I subsequently found

 =

So I decided to investigate all nine cases of the conversion of  to radicals, where ,  is one of sine, cosine, or tangent, and  is one of arcsine, arccosine, or arctangent. Our choice of  puts all three angles into the first quadrant, as shown in Table 1.

As per the Little Red Book entry, all nine cases considered in Table 2 must start with a conversion to exponential form. Then, the listed command(s) will complete the transition to radical form.

Surprisingly, not every case profits from the application of the rationalize command. Cases where rationalize is not indicated actually become worse if it is used. Other anomalies exist in Table 2. Generally, if a simplification command is not mentioned, it's because its use is either not helpful, or decidedly unhelpful.

The distillation from Table 2 should be a reminder that because simplification has no canonical form, the process of obtaining a particular form can be a challenge. The surprise for me was discovering that what worked for  did not work across-the-board for all the cases. I just happened to have listed in the Red Book one of the two cases that required rationalize.

In the previous blog, I pointed out that Maple's trigsubs command does not list the half-angle formulas for  or . I have since been delightfully surprised to discover that half-angles formulas for  are listed:

There are 14 formulas in the list , the last three of which are relevant. These three formulas do not involve radicals, as does the "obvious" formula

A formula with radicals will need intervention to pick the correct sign. The three formulas in  need no such consideration. The first two formulas in  and the one with the radical appear in my CRC Standard Mathematical Tables,  ed., Chemical Rubber Publishing Co., 1960, that I was given while taking freshman calculus, and still find useful even today. The third one in , immediate from the second, is useful in a manual derivation of some of the results in the last row of Table 2.

Table 3 summarizes a manual derivation of the contents of the first column in Table 2. If  then, by elementary trigonometry, , , , , and .

The last equalities in the first two rows of Table 3 illustrate a transformation that I can replicate only with brute force. For example, to write  as a multiple of , write , then form and solve the pair of equations  and . If this "algorithm" is applied to the radicals in the third column of Table 2, both  and  turn out to be binomial surds, not rational numbers.

Elementary trigonometry similar to what is seen in Table 3 will lead to the results in the remaining two columns of Table 2. So, the last item to examine is the conversion of the expressions  to exponential form. Just what happens to, say, , under this transformation?

Maple first uses the logarithmic equivalent of the arcsine function:

 =

then uses the exponential equivalent of the sine function:

 =

The composition of these two identities when  gives

 =

The conversion to radicals has taken place. What remains is the simplification of the radicals, and the verification that, in spite of the appearance of the complex unit , the expression is actually real.

These calculations illustrate one additional point, namely, that what takes place "under the hood" in Maple isn't necessarily how one would obtain the same result working with traditional manipulations. The elementary trigonometry captured in Table 3 just isn't the way Maple obtains the results in Table 2. Differences like these are what fill the pages of my Little Red Book of Maple Magic.
 

Download the worksheet: Converting_Half-Angl.mw