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Note: Not a duplicate of this.

So the so-called "special right triangles", or 30-60-90 and 45-45-90, are special triangles that have sines, cosines and tangents that can be calculated easily. Are these the only ones, and if so, why? Is it provable?

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Jack Pan
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    No, they're called "special right triangles" because their sines, cosines, and tangents, can be calculated easily. – Kenny Lau Jun 20 '16 at 01:57
  • Alright, I'll edit my question. – Jack Pan Jun 20 '16 at 01:58
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    You have to define "easily" so that your question makes sense –  Jun 20 '16 at 02:01
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    What do you mean by "easily"? I could arguably say $18^\circ-72^\circ-90^\circ$ and $36^\circ-54^\circ-90^\circ$ are special because they come up a lot in pentagons, but I certainly don't think the values of their $\sin$s and $\cos$s are easy to compute. However, people who are really good at trig and can do the proof very quickly likely disagree with me on that. – Noble Mushtak Jun 20 '16 at 02:01
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    It's like, we just defined those triangles to be special right triangles, so proving that no more exists doesn't make any sense to me. – Kenny Lau Jun 20 '16 at 02:01
  • Trig functions for some other triangles with nice angles can be computed fairly easily, like for the $15$-$75$-$90$. But these are not called special in school. – André Nicolas Jun 20 '16 at 02:05
  • These are the right triangles that can tile the plane without overlap through reflections in their edges. See Schwartz triangle, in the "Triangles for the Euclidean plane" bit. – Akiva Weinberger Jun 20 '16 at 04:24

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I think that all of those rectangle triangles that you qualify as "special" have one thing in common: the ratio of at least two of the sides is a whole number or even a rational number.

In the $30-60-90$ the ratio of the hypothenuse to the side adjacent to the $60$ degrees angle is $2$ and in the $45-45-90$ the ratio of the two sides adjacent to the $90$ degrees angle is $1$.

What you are then looking for are values of interior angles $\theta$ such that either the $\sin \theta$ or the $\cos \theta$ or both are rational numbers. This implies that there are an infinite number of those special triangles. The only thing is that they may not be whole numbers like $30$, $60$ or $45$ in degrees, but that does not matter since even those angles in radians are not whole numbers, not even rational.

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