Do they even teach the halfangle formulas in Trig anymore? They're not even in Mathematica 12.2! Macsyma had them forever. And they were useful. So here I was thinking the kids already knew them, and planning to ask them why the Trig books never mention "thirdangle" formulas, and why don't the kids derive them for themselves. But if they try to invert the triple angle formulas, they hit the casus irreducibilis bizarreness, where they need to solve a cubic whose real roots cannot be expressed in real terms. Kids should meet this while still young enough to struggle against it. And then they'd know why the Trig books never bring it up. But the books have an excuse: Bringing it up requires kids to already know complex numbers. Lots of prerequisites is not a good way to sell textbooks. But if the booksellers would accept the complex numbers prerequisite, instead of explaining about the casus pain-in-the-asus, they could actually provide a nifty thirdangle formula! Sin[x/3] == 1/2 Re[(1 + I Sqrt[3]) (I Cos[x] + Sin[x])^(1/3)], -π/2 ≤ x < 3π/2 E.g., In[212]:= %/. x -> π/6 Out[212]= Sin[π/18] == 1/2 Re[(1/2 + (I Sqrt[3])/2)^(1/3) (1 + I Sqrt[3])] In[213]:= %% /. x ->π/3 Out[213]= Sin[π/9] == 1/2 Re[(1 + I Sqrt[3]) (I/2 + Sqrt[3]/2)^(1/3)] In[214]:= %%%/. x -> π/4 (* Sin[π/12] == *) Out[214]= (-1 + Sqrt[3])/(2 Sqrt[2]) == 1/2 Re[((1 + I)^(1/3) (1 + I Sqrt[3]))/2^(1/6)] —rwg