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Suppose $3n$ is an angle between $0$ and $90$ degrees and that $n$ is an integer. By considering half squares and half equilateral triangles it's easy to obtain expressions for $\sin(3n)$ if $n=10$, $n=15$ and $n=20$. (By "expressions" I mean closed expressions involving addition, subtraction, multiplication, division and square roots.)

Using the addition/subtraction formulas for sine and cosine we can easily handle the cases $n=5$, and $n=25$ as well. I suppose that if I can find $\sin3$ then I can use these formulas to find $\sin(3n)$ for any $n$. Right? So my question is really this:

How do I find $\sin3$ (where the angle is measured in degrees)?

Ali Caglayan
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    https://math.la.asu.edu/~surgent/mat170/Exact_Trig_Values.pdf – Rocket Man Jun 06 '14 at 11:33
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    http://www.wolframalpha.com/input/?i=sin+3+deg – lhf Jun 06 '14 at 11:36
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    Either based on the pentagon (draw in the diagonals and use similar triangles) or via using trig identities to set up an equation you can solve, you can obtain the values of the trigonometric functions at $72^\circ$ (or $18^\circ$ or $36^\circ$ or $54^\circ$). Then $3^\circ = 75^\circ - 72^\circ$. –  Jun 06 '14 at 11:39
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    In terms of radians $3$ degrees is $\pi/60$, so an expression for the sine (and cosine) of $3$ degrees can be given in terms of cyclotomic roots of unity, which are solvable by expressions as radicals and rational operations. – hardmath Jun 06 '14 at 11:39
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    See http://en.wikipedia.org/wiki/Exact_trigonometric_constants#3.C2.B0:_regular_60-sided_polygon and http://www.intmath.com/blog/how-do-you-find-exact-values-for-the-sine-of-all-angles/6212. – lhf Jun 06 '14 at 11:40

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Here is a summary from this page:

$\sin 3^{\circ} = \sin (18^{\circ} – 15^{\circ})$, and you can use the addition formula.

$\sin (18^{\circ})$ can be found by the half-angle formula once you know $\sin (36^{\circ})$, which is related to the pentagon.

$\sin (15^{\circ})$ can be found by the half-angle formula with $\sin (30^{\circ})$, which you probably know.

lhf
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