Where do the radical expressions for the trig functions of various rational multiples of $pi$ come from?
So if you look on the Wikipedia page for "Trigonometric constants expressed as real radicals", you will get a boatload of formulas for the trig functions of various rational multiples of $pi$.
My question is, how were all these formulas deduced and/or derived? What is the proof that these formulas are correct? And are there any more potential formulas like this that can be derived, but not found by a mathematician?
trigonometry
edited Dec 9 at 14:52
Blue
47.6k870151
asked Dec 9 at 14:44
Xavier Stanton
330211
add a comment |
So if you look on the Wikipedia page for "Trigonometric constants expressed as real radicals", you will get a boatload of formulas for the trig functions of various rational multiples of $pi$.
My question is, how were all these formulas deduced and/or derived? What is the proof that these formulas are correct? And are there any more potential formulas like this that can be derived, but not found by a mathematician?
trigonometry
edited Dec 9 at 14:52
Blue
47.6k870151
asked Dec 9 at 14:44
Xavier Stanton
330211
2
Can you link to the page or write a few of the formulas down? Usually a brief explanation is given as to how to the formulas are derived; regarding formulas that might exist undiscovered...probably, but if nobody has found it, we couldn’t know.
– Clayton
Dec 9 at 14:47
2
The values are typically derived by taking the simplest cases (for, say, $pi/2$, $pi/3$, $pi/4$, and others that arise from various constructible regular polygons), hitting them with half-angle identities to get really small values, and then using multiple-angle identities. It's a pretty mechanical process, and more-or-less how trig tables were generated for centuries before we had calculators, so there's not much new to do in this space. However, finding "nice" forms and/or patterns, can be an interesting challenge. See this answer.
– Blue
Dec 9 at 14:59
You can see from the banner that the "page might contain original research".
– John Glenn
Dec 9 at 15:09
add a comment |
So if you look on the Wikipedia page for "Trigonometric constants expressed as real radicals", you will get a boatload of formulas for the trig functions of various rational multiples of $pi$.
My question is, how were all these formulas deduced and/or derived? What is the proof that these formulas are correct? And are there any more potential formulas like this that can be derived, but not found by a mathematician?
trigonometry
edited Dec 9 at 14:52
Blue
47.6k870151
asked Dec 9 at 14:44
Xavier Stanton
330211
So if you look on the Wikipedia page for "Trigonometric constants expressed as real radicals", you will get a boatload of formulas for the trig functions of various rational multiples of $pi$.
My question is, how were all these formulas deduced and/or derived? What is the proof that these formulas are correct? And are there any more potential formulas like this that can be derived, but not found by a mathematician?
trigonometry
trigonometry
edited Dec 9 at 14:52
Blue
47.6k870151
asked Dec 9 at 14:44
Xavier Stanton
330211
edited Dec 9 at 14:52
Blue
47.6k870151
asked Dec 9 at 14:44
Xavier Stanton
330211
edited Dec 9 at 14:52
Blue
47.6k870151
edited Dec 9 at 14:52
Blue
47.6k870151
edited Dec 9 at 14:52
Blue
47.6k870151
47.6k870151
asked Dec 9 at 14:44
Xavier Stanton
330211
asked Dec 9 at 14:44
Xavier Stanton
330211
asked Dec 9 at 14:44
Xavier Stanton
330211
330211
2
Can you link to the page or write a few of the formulas down? Usually a brief explanation is given as to how to the formulas are derived; regarding formulas that might exist undiscovered...probably, but if nobody has found it, we couldn’t know.
– Clayton
Dec 9 at 14:47
2
The values are typically derived by taking the simplest cases (for, say, $pi/2$, $pi/3$, $pi/4$, and others that arise from various constructible regular polygons), hitting them with half-angle identities to get really small values, and then using multiple-angle identities. It's a pretty mechanical process, and more-or-less how trig tables were generated for centuries before we had calculators, so there's not much new to do in this space. However, finding "nice" forms and/or patterns, can be an interesting challenge. See this answer.
– Blue
Dec 9 at 14:59
You can see from the banner that the "page might contain original research".
– John Glenn
Dec 9 at 15:09
add a comment |
2
Can you link to the page or write a few of the formulas down? Usually a brief explanation is given as to how to the formulas are derived; regarding formulas that might exist undiscovered...probably, but if nobody has found it, we couldn’t know.
– Clayton
Dec 9 at 14:47
2
The values are typically derived by taking the simplest cases (for, say, $pi/2$, $pi/3$, $pi/4$, and others that arise from various constructible regular polygons), hitting them with half-angle identities to get really small values, and then using multiple-angle identities. It's a pretty mechanical process, and more-or-less how trig tables were generated for centuries before we had calculators, so there's not much new to do in this space. However, finding "nice" forms and/or patterns, can be an interesting challenge. See this answer.
– Blue
Dec 9 at 14:59
You can see from the banner that the "page might contain original research".
– John Glenn
Dec 9 at 15:09
2
2
Can you link to the page or write a few of the formulas down? Usually a brief explanation is given as to how to the formulas are derived; regarding formulas that might exist undiscovered...probably, but if nobody has found it, we couldn’t know.
– Clayton
Dec 9 at 14:47
Can you link to the page or write a few of the formulas down? Usually a brief explanation is given as to how to the formulas are derived; regarding formulas that might exist undiscovered...probably, but if nobody has found it, we couldn’t know.
– Clayton
Dec 9 at 14:47
2
2
The values are typically derived by taking the simplest cases (for, say, $pi/2$, $pi/3$, $pi/4$, and others that arise from various constructible regular polygons), hitting them with half-angle identities to get really small values, and then using multiple-angle identities. It's a pretty mechanical process, and more-or-less how trig tables were generated for centuries before we had calculators, so there's not much new to do in this space. However, finding "nice" forms and/or patterns, can be an interesting challenge. See this answer.
– Blue
Dec 9 at 14:59
The values are typically derived by taking the simplest cases (for, say, $pi/2$, $pi/3$, $pi/4$, and others that arise from various constructible regular polygons), hitting them with half-angle identities to get really small values, and then using multiple-angle identities. It's a pretty mechanical process, and more-or-less how trig tables were generated for centuries before we had calculators, so there's not much new to do in this space. However, finding "nice" forms and/or patterns, can be an interesting challenge. See this answer.
– Blue
Dec 9 at 14:59
You can see from the banner that the "page might contain original research".
– John Glenn
Dec 9 at 15:09
You can see from the banner that the "page might contain original research".
– John Glenn
Dec 9 at 15:09
add a comment |
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2
Can you link to the page or write a few of the formulas down? Usually a brief explanation is given as to how to the formulas are derived; regarding formulas that might exist undiscovered...probably, but if nobody has found it, we couldn’t know.
– Clayton
Dec 9 at 14:47
2
The values are typically derived by taking the simplest cases (for, say, $pi/2$, $pi/3$, $pi/4$, and others that arise from various constructible regular polygons), hitting them with half-angle identities to get really small values, and then using multiple-angle identities. It's a pretty mechanical process, and more-or-less how trig tables were generated for centuries before we had calculators, so there's not much new to do in this space. However, finding "nice" forms and/or patterns, can be an interesting challenge. See this answer.
– Blue
Dec 9 at 14:59
You can see from the banner that the "page might contain original research".
– John Glenn
Dec 9 at 15:09