# Tricky trigonometrical triangle teaser

A classics teacher covering detention asked the question:

One angle of a triangle is ${96}^\circ$, find the other two angles $\psi, \theta$ given that:

$2\sin4\theta = 2\sin2\psi + 2\cos2\psi + 3\sec3\theta +2\alpha +2\sec2\psi$

Please find the other two angles so you can go home.

Addendum: The teacher knows basic arithmetic and geometry, but trigonometry is Greek to him. His proudest mathematical moment occurred during an interview when asked to find the value of,

$3\sin3\psi \cdot 3 \tan3\theta$ if $\psi={30}^\circ$ and $\theta={15}^\circ$

answering even before the interviewer had finished speaking. He got the job. He said confidently

9

Hint

$4\cosh3\psi + 3\sinh4\theta$

• @JeffZeitlin: It's an enigmatic puzzle. Since when do classics teachers pose maths problems? – M Oehm Aug 25 '17 at 13:49
• @JeffZeitlin That may be what it looks like, but that doesn't make it true. – dcfyj Aug 25 '17 at 13:54
• The point Oehm is making is that the tag is enigmatic. So there's more to it than the math question. – n_plum Aug 25 '17 at 13:54
• There's definitely something funny going on here: look at that $2\alpha$. – Gareth McCaughan Aug 25 '17 at 13:55
• There's too many variables to solve uniquely - the equation needs to be interpreted as a puzzle. – Tom Aug 25 '17 at 14:14

Aha! The last hint was very informative.

The classics teacher ...

... doesn't know much about trigonometry, but he certainly knows his Greek letters. He sees the equation as number–word pairs. The Greek letters are spelled out and all those arcane operators are just decoration.

These number–word pairs form single letters: The number defines which letter to extract from the word.

So the last hint ...

... spells out hint:

4 cosh 3 psi + 3 sinh 4 theta

The first hint ...

... means nine, which is in accordance with the teacher's answer:

3 sin 3 psi · 3 tan 3 theta

The answer to the original question is:

It (is) isosceles.

2 sin 4 theta = 2 sin 2 psi + 2 cos 2 psi + 3 sec 3 theta + 2 alpha + 2 sec 2 psi

So the both remaining angles of the triangle ...

... are both 42°. And if that isn't a nice answer, I don't know.

• Perhaps the angle in degrees would have rendered more consistantly as: 3 cos 2π + 3 max(3 cot 3ψ, 4γ + 2 sec 2ψ, 3 sin 3ψ + 3 tan 3ϑ. – M Oehm Nov 18 '17 at 13:01
• $3\tan3\psi + 3\sec3\theta$ :) – Tom Nov 18 '17 at 19:14
• I had this puzzle in my favorite and I'm just seeing this now : nice puzzle and answers – Fabich Feb 24 '18 at 21:08

Well, I guess

The L.H.S is an even number. So the R.H.S also must be an even number. That gives a hint that 3 sec3θ is even. Thereby giving Sec 3θ = 4/3 or Cos 3θ = 3/4 or θ = 41.4 /3 = 13.8 and ψ = 70.2

• The LHS is not always even. If $\theta=7.5^\circ$, then LHS = 1 is odd. – Mike Earnest Aug 26 '17 at 0:44