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As the title explains, I am looking to find some more difficult trigonometric equations from past olympiads or contests. A common one is $\tan 2x \tan \left(2x +\frac{\pi}{3}\right) \tan \left(2x +\frac{2\pi}{3}\right)=\sqrt{3}$, so if anyone could find some interesting ones for me to have a go at with a similar level of difficulty, I'd appreciate it.

pjs36
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wrb98
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4 Answers4

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find the roots of $$ x^6 + x^5 - 5 x^4 - 4 x^3 + 6 x^2 + 3 x - 1 $$

Not obvious about the trigonometry. I wrote up an answer, anyone who wishes to see that email me and ask for my problem_2171819.pdf

enter image description here

Will Jagy
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  • Uh ... all the roots have residue $2\bmod 13$. And numbers of the form$\cos(2k\pi/13)$ have residue $1$ ... . – Oscar Lanzi Jan 06 '24 at 03:08
  • @OscarLanzi right, page 12 in Reuschle https://archive.org/details/tafelncomplexer00unkngoog/page/12/mode/2up – Will Jagy Jan 06 '24 at 03:43
  • I would try setting $x=2\cos\theta$ (the 2 comes from the modulo 13 residues of the roots) and then using this method, with $\sin\theta$ as the multiplier. (You might have guessed I know the answer.) – Oscar Lanzi Jan 06 '24 at 12:19
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Many readers know that the sine of $18°$ or $\pi/10$ radians is $(\sqrt5-1)/4$. But given only the number $(\sqrt5-1)/4$, find its inverse sine without having memorized the trigonometric fact. You may use the fact that the answer $\theta$ satisfies $4\sin^2\theta+2\sin\theta-1=0$, from the quadratic equation for $(\sqrt5-1)/4$. Either degrees or radians is accepted as units for the answer.

I won't reveal my entire solution, just a summary. (1) Multiply the quadratic equation by $\cos\theta$. (2) Use the trigonometric sum-product relations to render $4\sin^2\theta\cos\theta$ and $2\sin\theta\cos\theta$ in terms of $\sin k\theta$ or $\cos k\theta$ for whole numbers $k$. (3) Substitute these results into the equation from (1) and observe that terms cancel, leaving $\sin m\theta=\cos n\theta$ for some $m$ amd $n$. (4) Conclude that $m\theta$ and $n\theta$ are complementary, thus solve for $\theta$.

Oscar Lanzi
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Let $\alpha, \beta, \gamma$ be the angles between the horizontal for the figure below.

enter image description here

Find $\alpha + \beta + \gamma$

  1. Find $\sin^2 1^\circ + \sin^2 2^\circ + \cdots + \sin^2 90^\circ$

  2. Show that $\sin 1^\circ + \sin 2^\circ + \cdots + \sin 90^\circ = \frac {\sin 0.5 + \sin 90.5^\circ}{2\sin 0.5^\circ}$

user317176
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I googled "math olympiad trigonometry" and came up with the following:

here: $tan^220+tan^240+tan^280=33$

here: $A + B + C$ is an integral multiple of $\pi$. $x, y,$ and $z$ are real numbers. If $x\sin(A)+y\sin(B)+z\sin(C)=x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0$, show that $x^n\sin(nA)+y^n \sin(nB) +z^n \sin(nC)=0$ for any positive integer $n$.

Plus two more from same site.

Here is one from math stack exchange: Help me solve this olympiad challenge?

And so forth.

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