Exact trig. values for 3, 6, 9, etc. degrees

[Dave L. Renfro]

In case anyone is interested in radical expressions for trig.
functions at 3, 6, 9, 12, etc. degrees, see the following:

1. Search in this book for "Gelin", choose "Page 1".
Look at 3'rd to 13'th pages beyond this. Or, if you
download the volume, it's pp. 338-348 of the .pdf file.
http://books.google.com/books?id=5r0KAAAAIAAJ

[Gelin previously published in the journal Mathesis
Recueil Mathematique (Volume 8, 1888, Supplement 3)
a list of values, with rationalized denominators and
in "simple" factored form, for the sine, tangent, and
secant of 3, 6, 9, ..., 90 degree angles. However, this
earlier volume doesn't seem to be on the internet yet.]

2. Search in this book for "every third degree" (include quotes),
choose Page xxxix. Or, if you download the volume, look at
pp. 48-49 of the .pdf file.
http://books.google.com/books?id=rHcAAAAAMAAJ

[The same table just above is also in the later editions
of Hutton's "Mathematical Tables". I don't know when
it first appeared, but I do know it's not in the 1811 edition.]

3. Search in this book for "in the following table" (include
quotes), choose Page 72. Or, if you download the volume,
look at p. 93 of the .pdf file.
http://books.google.com/books?id=_ktLAAAAMAAJ

4. I'm not sure, but I believe Johann Heinrich Lambert was
the first person who published exact radical values for the
sine of 3, 6, 9, etc. degree angles. A table of values is in
Volume 1 of his "Collected Works". The table is in an item
that was published in 1770. I can't seem to find Lambert's
"Collected Works" on the internet, but the local university
library where I live has the 2-volumes of his "Collected
Works".

Dave L. Renfro

[Dave L. Renfro]

Dave L. Renfro wrote:

[snip]

Here are two more references for this topic:

5. Search in this book for "nombres naturels" (include quotes),
choose Page 58. Or, if you download the volume, look at
pp. 84-85 of the .pdf file.
http://books.google.com/books?id=1uZJAAAAMAAJ

6. Search in this book for "Thomas Simpson" (include quotes),
choose Page 403 {for the sine of 9, 15, 18, 22.5, 27, etc.
degrees} and search in this book for "tangentes et
cotangentes" (include quotes), choose Page 419 {for
the tangent of 9, 15, 18, 22.5, 27, etc. degrees}.
Or, if you download the volume, look at pp. 630 & 638-639
of the .pdf file.
http://books.google.com/books?id=JvAIAAAAQAAJ

Dave L. Renfro