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$\begingroup$

For example:

  1. $\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$
  2. $\sin(18^\circ) = \frac{\sqrt{5}}{4} - \frac{1}{4}$
  3. $\sin(30^\circ) = \frac{1}{2}$
  4. $\sin(45^\circ) = \frac{1}{\sqrt{2}}$
  5. $\sin(67 \frac{1}{2}^\circ) = \sqrt{ \frac{\sqrt{2}}{4} + \frac{1}{2} }$
  6. $\sin(72^\circ) = \sqrt{ \frac{\sqrt{5}}{8} + \frac{5}{8} }$
  7. $\sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$
  8. ?

Is there is a list of known exact values of $\boldsymbol \sin$ somewhere?

Found a related post here.

  • 4
    See [this](http://mathworld.wolfram.com/TrigonometryAngles.html).2012-07-30
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    Since we have the half angle formula, the list must be infinite... And I am guessing you are asking which values can be expressed as an algebraic number, since all values are known analytically using the formula $\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(1n+1)!}$.2012-07-30
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    @copper.hat: Yes, algebraically is what I am looking for, not approximately or a series.2012-07-30
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    @J.M. So the answer is for all $\frac{m}{n} \pi$ where $m$ and $n$ are non-zero integers? Can you produce an answer for $67.5^\circ = \frac{3}{8} \pi$ for example and I will accept it.2012-07-30
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    @ja72 : $\sin(3 \pi/8) = \sqrt{2+\sqrt{2}}/2$2012-07-30
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    @ja72 While $\sin(m\pi/n)$ is _algebraic_ for all integer $m$ and $n$, that doesn't mean it's expressable in terms of radicals for all those values. You may find the notion of _constructibility_ ( http://en.wikipedia.org/wiki/Constructible_polygon )interesting, though that doesn't cover the case of other radicals; for instance, the value of $\sin(\pi/7)$ can be expressed with cube roots (but not with cube roots or real numbers). Have a look at http://en.wikipedia.org/wiki/Root_of_unity for more information on that case...2012-07-30
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    Actually there are others besides $m \pi/n$, e.g. $\sin(\arctan(1/2))=1/\sqrt{5}$. You may find this a bit of a cheat though, since the angle is specified using inverse trig functions.2012-07-30
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    If $x/\pi$ is an algebraic irrational, $\sin(x)$ is transcendental by the Gelfond-Schneider theorem.2012-07-30
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    A complete list would be obtained by taking arcsin values on the set of all algebraic numbers. Of course, it's rather unlikely that the resulting numbers will be algebraic with respect to $\pi$... You should make it clear what you expect from the angles and from the sine expressions.2012-07-30
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    In what sense do we know the value of $\sqrt{2}$ exactly?2012-07-30
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    @QiaochuYuan: Good comment, so I guess we need to draw the line at transcendent functions.2012-07-30

5 Answers 5

-2

Algebraically exact value of Sine for all integer angles is possible. Please visit https://archive.org/details/ExactTrigonometryTableForAllAnglesFinal for the list of exact values for Sine of integer angles in degrees.

  • 0
    Interesting idea, but the list is kind of useless due to bad math formatting. The easiest way to do $LaTeX$ is to download `LyX` and `MikTeX`. Also url link only answers are frowned upon here. Maybe you can do a short example of how to derive the results of one angle so the reader can understand better the process.2016-11-06
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    @ja72 May you assist me to copy edit that document? I have not idea to use all of these. I cannot format as online forum requires formatting the mathematics. ** If someone assisted** me for copy edit on Precise-Rewritten method (and other new methods), every scholar may know the new idea for **new method for exact trigonometric values**. My (Breaking Classical Rules in Trigonometry- Mission 2050) un-skill on mathematics formating discouraging me to expose all of those new idea.2016-11-07
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    Place math expressions between dollar signs `$1 + x^2$` → $1+x^2$ see http://math.stackexchange.com/help/notation. To make math in 'display' form (in its own paragraph) use double dollar signs `$$1+\frac{1}{x}$$` →$$1+\frac{1}{x}$$ Find other posts and right-click on math and select view math as TeX to see what commands make different parts. Like `\frac` or `\sin` or `\sqrt`2016-11-07
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    Also download `LyX` (free, full-installer) https://www.lyx.org/Download and work on your own with generating LaTeX documents instead of using Word.2016-11-07
12

$\sin 3^\circ=\frac{(\sqrt{3}+1) (\sqrt{5}-1)}{8 \sqrt{2}}-\frac{(\sqrt{3}-1) \sqrt{5+\sqrt{5}}}{8}$.

Solving a cubic equation you can get a huge expression for $\sin 1^\circ$ in radicals, and therefore, for any $\sin n^\circ$.

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    Can you show your work for $\sin 3^\circ$?2012-07-30
  • 1
    @ja72: Note that $3^\circ=18^\circ-15\circ$. Now use the formula $\sin(a-b)=\sin a\cos b-\cos a\sin b$. In your post you mentioned expressions for the sines of $18^\circ$ and $15^\circ$. From those and the Pythagorean Theorem you get their cosines.2012-07-30
  • 0
    But solving the cubic equation for $\sin 1^\circ$ will require taking the cube root of a complex number, won't it?2012-07-30
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    @RahulNarain: Yes, you are right.2012-07-30
  • 0
    The huge expression for $\sin(1^\circ)$ can be found [here](http://math.stackexchange.com/q/94478/19341)...2012-07-30
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    It's really not so bad if you don't try to put it in one expression. Here's how Maple can optimize it. $$\eqalign{t_1 &= 5^{1/2}\cr t_4 &= (30+6 t_1)^{1/2}\cr t_9 &= (2 t_4(1-t_1)+28-4 t_1)^{1/2}\cr t_{11} &= (-1+t_1+t_4+i t_9)^{1/3}\cr t_{13} &= t_{11}^2\cr t_{22} &= (128+32 t_{11}+2 t_4 t_{13}+2 t_1 t_{13}-2 t_{13}- 2 i t_9 t_{13})^{1/2}\cr t_{25} &= (32-2 t_{22})^{1/2}\cr \text{result} &= t_{25}/8 \cr}$$2012-07-31
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    @RobertIsrael +1 nice. Do the indices $k$ of $t_k$ mean anything?2012-07-31
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    @RahulNarain I think the mentinoned cube root can be found in Robert's $t_{11}$.2012-07-31
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    @draks: No, they don't mean anything particular, it's just an artefact of the optimization process. Presumably Maple ran through at least 25 intermediate variables, and winnowed it down to 7.2012-07-31
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We use radian notation. Every rational multiple of $\pi$ has trigonometric functions that can be expressed using the ordinary arithmetic operations, plus $n$-th roots for suitable $n$.

This is almost immediate if we use complex numbers, since $(\cos(2\pi/n)+i\sin(2\pi/n)^n=1$.

But it is known, for example, that there is no expression for $\sin(\pi/9)$ that starts from the integers, and uses only the ordinary operations of arithmetic and roots in which every component is real.

The following more restricted problem has a long history because of its close connection with the problem of which angles are constructible by straightedge and compass.

Let $\theta=\frac{m}{n}\pi$, where $m$ and $n$ are relatively prime. Restrict our algebraic operations to the ordinary operations of arithmetic, plus square roots only, The trigonometric functions of $\theta$ are so expressible iff $n$ has the form $$n=2^k p_1p_2\cdots p_s,$$ where the $p_i$ are distinct Fermat primes.

A Fermat prime is a prime of the form $2^{\left(2^t\right)}+1$. There are only five Fermat primes known: $3$, $5$, $17$, $257$, and $65537$. It is not known whether or not there are more than five.

  • 0
    @tomasz: Thanks for the suggestion. Done.2012-07-30
  • 0
    +1 - I used this no problem for $\pi/7$ and $\pi/9$. But, I don't follow your comment that every rational multiple of $\pi$ has trig functions expressible in $n$th roots for suitable $n$. We could certainly express them in terms of roots of a polynomial system obtained by expanding out the $\text{cis}(2pi/n)^n$. If we can solve that in terms of roots, then you're correct. I obtained an irreducible 10th degree polynomial for $\pi/11$, however.2016-01-26
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    I was using roots in the sense of the radicals of the problem about the quintic, which automatically makes all solvable by radicals, namely roots of unity. The problem changes if we insist on real radicals.2016-01-26
  • 0
    That makes sense - thanks!2016-01-26
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    @AndréNicolas But there are some quintic and above polynomials that can't even be solved using roots of unity, are there not? For example, the wikipedia article on quintic mentions that in order to solve the quintic generally, one needs to use bring radicals, or something similar -- not simply roots of unity. Is there some reason why these polynomials would never show up when we look at sin and cos of rational multiples of $pi$?2016-04-04
  • 0
    That has been (admittedly quickly) dealt with in the answer. The cosine of $2\pi/n$ is $1/2$ the sum of two $n$-th roots of unity. So in the classical sense of expressibility by radicals, it can be done. That is not the case for some quintics (and higher) with rational coefficients.2016-04-04
5

In January 2008 I posted several references published in the 1800s of tables that give exact values for the sine and cosine of $3$, $6$, $9$, …, $90$ degree angles. (Among the integer degree angles, only those that are multiples of $3$ can be expressed in real-radical form.) See Math Forum archive for 1st post and Math Forum archive for 2nd post.

The best table I know of was prepared by the Belgium mathematician E. Gelin in the 1880s. His table gives a list of values, with rationalized denominators, for all six trig. functions evaluated at $3$, $6$, $9$, …, $90$ degree angles. I know of three places where his table has been published:

Mathesis Recueil Mathematique (1) 8 (1888), Supplement 3. [See pp. 327-333 of the downloaded .pdf file.]

Mathesis Recueil Mathematique (3) 6 (1906), Supplement 3. [See pp. 338-348 of the downloaded .pdf file.]

E. Gelin, Éléments de Trigonométrie Plane et Sphérique (1888). [See pp. 59-62, which is equivalent to pp. 66-69 of the downloaded .pdf file.]

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. Lambert’s table was reprinted two or three times in the first half of the 1800s (e.g. one was in Crelle’s Journal [= Journal für die reine und angewandte Mathematik]), but I don’t have the exact references with me now.

5

Starting with $\tan(\pi/3)=\sqrt{3}$ and $\tan(\pi/4)=1$ and using $$ \tan(x/2)=\frac{\sqrt{1+\tan^2(x)}-1}{\tan(x)}\tag{1} $$ and $$ \tan(x+y)=\frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}\tag{2} $$ and $$ (\cos(x),\sin(x))=\frac{(1,\tan(x))}{\sqrt{1+\tan^2(x)}}\tag{3} $$ we can construct the sine and cosine of all rational multiples of $\pi$ where the denominator is a power of $2$, or $3$ times a power of $2$.

For example, $x=\pi/4$ with $(1)$ gives $$ \tan(\pi/8)=\sqrt{2}-1\tag{4} $$ then $(2)$ and $(4)$ yields $$ \begin{align} \tan(3\pi/8) &=\tan(\pi/4+\pi/8)\\ &=\frac{1+\tan(\pi/8)}{1-\tan(\pi/8)}\\ &=\sqrt{2}+1\tag{5} \end{align} $$ Then $(3)$ and $(5)$ give $$ (\cos(3\pi/8),\sin(3\pi/8))=\frac{(1,\sqrt{2}+1)}{\sqrt{4+2\sqrt{2}}}\tag{6} $$