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There are plenty of formulae for half-angles, double angles, but I haven't manage to find nor derive an expression for $\cos\left(\frac{\theta}{3}\right)$ in terms of $\cos(\theta)$. How would one proceed?

Extending the question, is there a general way to express $\cos\left(\frac{\theta}{k}\right)$ in terms of $\cos(\theta)$? (with $k\in\mathbb{N}$).

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    @XanderHenderson Maybe you have misead $\pi$ for $\theta$? Please check again your proposal for closure. – user Nov 28 '19 at 22:03
  • @the_candyman, sorry I did not mean to use $x$, it must be late... –  Nov 28 '19 at 22:04
  • @XanderHenderson, I don't think it is a duplicate, I am looking for general $\theta$ and not only $\pi$. –  Nov 28 '19 at 22:05
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    No, @user, I have not. The linked question asks for one to find a formula for $\sin(3a)$ in terms of $\cos(a)$ and $\sin(a)$, then to use that formula to obtain the value of $\sin(\pi/3)$. But the method for obtaining the formula $\sin(3a)$ is nearly identical to that for $\cos(3a)$, which is what is required here. – Xander Henderson Nov 28 '19 at 22:05
  • @XanderHenderson To me, it seems not a duplicate! Anyway you are of course more expert than me in closure! – user Nov 28 '19 at 22:09
  • There is also this question, which seems to address the current (after editing) version of the current question. – Xander Henderson Nov 28 '19 at 22:24
  • @XanderHenderson I'm sure I've already seen something more specific and strictly related. Maybe it has been deleted? I'll take a look for it. – user Nov 28 '19 at 22:26
  • And there is also this answer, which discusses the derivation of triple angle formulae. – Xander Henderson Nov 28 '19 at 22:33
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    @JoséCarlosSantos It is not strictly a duplicate, maybe it covers also this one but I wouldn't classify it as a duplicate. I'm sure there is some other more fitting question. – user Nov 28 '19 at 22:35
  • @JoséCarlosSantos What about this one? This also seems cover this OP. – user Nov 28 '19 at 22:42
  • @user Note that the first answer provides a complete answer to this question. – José Carlos Santos Nov 28 '19 at 22:45
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    @JoséCarlosSantos I noticed that. Indeed I claim that it covers it but it is not a duplicate. Moreover note that the asker here is asking also for a more general extension: "is there a general way to express $\cos\left(\frac{\theta}{k}\right)$ in terms of $\cos(\theta)$? (with $k\in\mathbb{N}$)". – user Nov 28 '19 at 22:49

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To express $\cos\left(\frac{\theta}{3}\right)$ in terms of $\cos \theta$ we can use Triple-angle formulae

$$\cos (3\theta) = 4 \cos^3\theta - 3 \cos\theta$$

but we need to solve a cubic equation.

Refer also to

Edit

For the more general issue we can refer to Chebyshev method

$$\cos(n\theta ) = 2 \cos \theta \cdot \cos((n-1)\theta) − \cos((n-2)\theta)$$

user
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