MaplePrimes - comments on Post, Trigonometric functions in radicals

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    <title>MaplePrimes - comments on Post, Trigonometric functions in radicals</title>
    <link>http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals</link>
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    <pubDate>Tue, 09 Jun 2026 15:07:52 GMT</pubDate>
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    <description>The latest comments added to the Post, Trigonometric functions in radicals</description>
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      <title>MaplePrimes - comments on Post, Trigonometric functions in radicals</title>
      <link>http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals</link>
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      <title>It could be any integer number of degrees</title>
      <link>http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals?ref=Feed:MaplePrimes:Trigonometric functions in radicals:Comments#comment144207</link>
      <itunes:summary>&lt;p&gt;"... if the number of degrees is divisible by three" seems like a rather weak statement! Here I prove that the trigonometric functions for any integer number of degrees can be expressed in radicals. Below, I derive cos(20&amp;deg;) in radicals (because the expression is relatively simple) and cos(1&amp;deg;). Clearly, once you have cos(1&amp;deg;) in radicals you can get any trigonometric function of any integer number of degrees in radicals by applying secondary-school-level identities.&lt;/p&gt;
&lt;p&gt;I also used my font chart from my previous post to find that the degree symbol is character number 176, so I use that below.&lt;/p&gt;
&lt;p&gt;It is surprising that &lt;strong&gt;convert(..., radical) &lt;/strong&gt;does not apply these techniques. It seems limited to multiple-of-three degrees. &lt;br&gt; &lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #78000e; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt;restart;&lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #78000e; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt;&amp;deg;:= Pi/180:&lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #78000e; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt;cos(3*`20&amp;deg;`) = cos(60*&amp;deg;);&lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -16;" src="/view.aspx?sf=144207/455309/01c7d2afaca448a1c05930ef2f7e18e4.gif" alt="cos(3*`20&amp;deg;`) = 1/2" width="102" height="42"&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #78000e; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt;expand(%);&lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -16;" src="/view.aspx?sf=144207/455309/55b6635b7c164e1438455fd135c733fb.gif" alt="4*cos(`20&amp;deg;`)^3-3*cos(`20&amp;deg;`) = 1/2" width="191" height="42"&gt;&lt;/p&gt;
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&lt;td&gt;&lt;span style="color: #78000e; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #78000e; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt;solve({%, cos(`20&amp;deg;`) &amp;gt; 0}, cos(`20&amp;deg;`), Explicit);&lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -24;" src="/view.aspx?sf=144207/455309/0c8587772a3a78ecc66f5ce9a7d7d000.gif" alt="{cos(`20&amp;deg;`) = (1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)}" width="346" height="50"&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #78000e; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt;expand(cos(3*`1&amp;deg;`)) = convert(cos(3*&amp;deg;), radical);&lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -52;" src="/view.aspx?sf=144207/455309/781ce6cc60b901cb8542e8d222e8eb18.gif" alt="4*cos(`1&amp;deg;`)^3-3*cos(`1&amp;deg;`) = (-(1/16)*2^(1/2)+(1/8)*(5+5^(1/2))^(1/2)+(1/16)*2^(1/2)*5^(1/2))*3^(1/2)+(1/8)*(5+5^(1/2))^(1/2)+(1/16)*2^(1/2)-(1/16)*2^(1/2)*5^(1/2)" width="480" height="78" align="middle"&gt;&lt;/p&gt;
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&lt;td&gt;&lt;span style="color: #78000e; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #78000e; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt;solve({%, cos(`1&amp;deg;`) &amp;gt; 0}, cos(`1&amp;deg;`), Explicit);&lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -341;" src="/view.aspx?sf=144207/455309/17f6d5bdc85597c2180dc7e8d39bc7fe.gif" alt="{cos(`1&amp;deg;`) = (1/8)*(4*3^(1/2)*2^(1/2)*5^(1/2)-4*2^(1/2)*5^(1/2)-4*2^(1/2)*3^(1/2)+8*3^(1/2)*(5+5^(1/2))^(1/2)+4*2^(1/2)+8*(5+5^(1/2))^(1/2)+(8*I)*(32-2*5^(1/2)*2^(1/2)*(5+5^(1/2))^(1/2)-4*5^(1/2)*3^(1/2)+2*2^(1/2)*(5+5^(1/2))^(1/2)-4*3^(1/2))^(1/2))^(1/3)+2/(4*3^(1/2)*2^(1/2)*5^(1/2)-4*2^(1/2)*5^(1/2)-4*2^(1/2)*3^(1/2)+8*3^(1/2)*(5+5^(1/2))^(1/2)+4*2^(1/2)+8*(5+5^(1/2))^(1/2)+(8*I)*(32-2*5^(1/2)*2^(1/2)*(5+5^(1/2))^(1/2)-4*5^(1/2)*3^(1/2)+2*2^(1/2)*(5+5^(1/2))^(1/2)-4*3^(1/2))^(1/2))^(1/3)}" width="480" height="308" align="middle"&gt;&lt;/p&gt;
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&lt;p&gt;&lt;a href="/view.aspx?sf=144207/455309/cosine_1_degree.mw"&gt;Download cosine_1_degree.mw&lt;/a&gt;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, Trigonometric functions in radicals</description>
      <guid>144207</guid>
      <pubDate>Tue, 05 Mar 2013 22:51:24 Z</pubDate>
      <itunes:author>Carl Love</itunes:author>
      <author>Carl Love</author>
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      <title>Without I</title>
      <link>http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals?ref=Feed:MaplePrimes:Trigonometric functions in radicals:Comments#comment144210</link>
      <itunes:summary>&lt;p&gt;&lt;a href="http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals#comment144207"&gt;@Carl Love&lt;/a&gt; Is it possible to express cos(Pi/180) in radicals without the imaginary unit I?&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, Trigonometric functions in radicals</description>
      <guid>144210</guid>
      <pubDate>Tue, 05 Mar 2013 23:35:59 Z</pubDate>
      <itunes:author>Markiyan Hirnyk</itunes:author>
      <author>Markiyan Hirnyk</author>
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      <title>I doubt it</title>
      <link>http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals?ref=Feed:MaplePrimes:Trigonometric functions in radicals:Comments#comment144211</link>
      <itunes:summary>&lt;p&gt;I doubt that it could be expressed in radical form without the &lt;em&gt;I&lt;/em&gt;, because it comes from solving a cubic. Attempting to use &lt;strong&gt;evalc&lt;/strong&gt; and &lt;strong&gt;simplify&lt;/strong&gt; puts it back into a trigonometric form.&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, Trigonometric functions in radicals</description>
      <guid>144211</guid>
      <pubDate>Tue, 05 Mar 2013 23:56:56 Z</pubDate>
      <itunes:author>Carl Love</itunes:author>
      <author>Carl Love</author>
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      <title>Without complex numbers</title>
      <link>http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals?ref=Feed:MaplePrimes:Trigonometric functions in radicals:Comments#comment144212</link>
      <itunes:summary>&lt;p&gt;&lt;a href="http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals#comment144207"&gt;@Carl Love&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;I mean only real radicals without complex numbers.&lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, Trigonometric functions in radicals</description>
      <guid>144212</guid>
      <pubDate>Tue, 05 Mar 2013 23:57:35 Z</pubDate>
      <itunes:author>Kitonum</itunes:author>
      <author>Kitonum</author>
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      <title>trig ---&gt; radicals</title>
      <link>http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals?ref=Feed:MaplePrimes:Trigonometric functions in radicals:Comments#comment144213</link>
      <itunes:summary>&lt;p&gt;I remember, that Carl had some code at his Yahoo group&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, Trigonometric functions in radicals</description>
      <guid>144213</guid>
      <pubDate>Wed, 06 Mar 2013 02:32:44 Z</pubDate>
      <itunes:author>Axel Vogt</itunes:author>
      <author>Axel Vogt</author>
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      <title>When is cos(x) representible in radicals?</title>
      <link>http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals?ref=Feed:MaplePrimes:Trigonometric functions in radicals:Comments#comment144323</link>
      <itunes:summary>&lt;p&gt;&lt;a href="http://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals#comment144213"&gt;@Axel Vogt&lt;/a&gt;&amp;nbsp;&lt;/p&gt;
&lt;p&gt;Yes, I recall writing something like that, but I don't recall posting it. If you saw it, then I did. Someday soon, I'll have to go download all that Yahoo Maple stuff. I haven't looked at it in many years.&lt;/p&gt;
&lt;p&gt;The gist of code in question, IIRC, is that if&lt;/p&gt;
&lt;p&gt;&lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=bdafe547664eff4c683ad63f66f21d6a.gif" alt="theta=p*Pi/q/2^m/3^n"&gt;&lt;/p&gt;
&lt;p&gt;for integers &lt;em&gt;p&lt;/em&gt;,&lt;em&gt; q&lt;/em&gt;, &lt;em&gt;m&lt;/em&gt;, &lt;em&gt;n&lt;/em&gt; with&lt;em&gt; q&lt;/em&gt; = 1, 5, 7, or 11, then the trig functions of &lt;em&gt;&amp;theta;&lt;/em&gt; can be expressed in (complex) radicals. I can't recall if my code handled every case.&lt;/p&gt;
&lt;p&gt;The case &lt;em&gt;q&lt;/em&gt; = 11 is quite interesting because it involves solving a quintic. I wonder if there are higher values of &lt;em&gt;q&lt;/em&gt; for which the polynomial is solvable (even though Maple can't solve it). Does anyone here know? For &lt;em&gt;q&lt;/em&gt; odd, the polynomial to solve for &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=48deae877ab0d5413f0b648404784ed5.gif" alt="cos(Pi/2/q)"&gt; &lt;em&gt;&lt;/em&gt;is of the form &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=f5f9eecd50a985eba818e2ec5b8e405c.gif" alt="x*p(x^2)"&gt; where degree(&lt;em&gt;p&lt;/em&gt;) = (&lt;em&gt;q&lt;/em&gt;-1)/2 and all the roots are real and in (0,1).&lt;/p&gt;
&lt;p&gt;What if &lt;em&gt;q&lt;/em&gt; is a Fermat prime? For &lt;em&gt;q&lt;/em&gt;=17, that polynomial is degree 8. Maple should be able to compute the galois group, but I don't know how to interpret the results. Seems like there should be a connection between this and Gauss's proof of the compass-and-straight-edge constructability of a regular &lt;em&gt;n&lt;/em&gt;-gon when &lt;em&gt;n&lt;/em&gt; is a Fermat prime. (Gauss's wanted his tombstone to be engraved with a regular 17-gon.)&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, Trigonometric functions in radicals</description>
      <guid>144323</guid>
      <pubDate>Fri, 08 Mar 2013 02:51:14 Z</pubDate>
      <itunes:author>Carl Love</itunes:author>
      <author>Carl Love</author>
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