# Identifying Trigonometrical proofs

How can we identify trigonometrical proofs from geometrical proofs, do we have purely trigonometrical proof of Pythagoras theorem as claimed by two high school students ?
https://www.scientificamerican.com/article/2-high-school-students-prove-pythagorean-theorem-heres-what-that-means/
As mathematics teachers we need to encourage students to find alternative proofs and for that purpose I think it's important to let them discuss positive and negative points of new approaches. I have following issues related to the high school students achievement.

1. How can you apply above mentioned proof for right angled isosceles triangles?
2. Can you say that approach is trigonometrical because of using trigonometrical formula $$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$$ since it also can be derived geometrically by right angled triangles?
3. Is it not possible to complete the proof by just adding the squares of two perpendicular sides without using the formula $$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$$ and say that proof is geometrical?
Here my question is related to how we have to update students using new developments in mathematics, therefore I hope you will not misunderstand this as a mathematics question.

https://youtu.be/p6j2nZKwf20?si=R-G00I5vyKsFalba
• Seems like more of a math question than a teaching one. And no, that students ask sbout it or you want teachers to opine in it doesn't change that. Commented Oct 21, 2023 at 18:23
• Fwiw, I think this is very unimportant versus actual teaching and learning. Beware diverting so much, like with history, from the actual course materials. Commented Oct 21, 2023 at 18:24
• @Raciquel thanks for your interest and giving opportunity to add further references. Now you can have better idea of what I meant. Commented Oct 22, 2023 at 1:46
• @guesttroll thanks for helping me to have better understanding by sharing your knowledge. I think only course materials not sufficient when it comes to the point of encouraging students to explore the subject and go for their own approaches without limiting them to a fixed frame. In this relevant post I'm talking about one of the different approaches of two high school students. I think this kind of awareness should be given to students to get inspired and to improve their investigative skills. Commented Oct 22, 2023 at 4:46

The proof: https://youtu.be/p6j2nZKwf20

For context, here's the main idea of the proof. Using the definition of sine, we have $$c^2 = \dfrac{2ab}{\sin 2\alpha}.$$ Our goal is to show that the RHS is equivalent to $$a^2 + b^2.$$ If $$a=b,$$ then this is obvious since $$\alpha = 45^\circ.$$ Otherwise, if $$a \neq b,$$ then we perform a geometric construction called the "waffle cone" and do some hairy algebra to get to the desired result (this forms the bulk of the proof).

1. How can you apply above mentioned proof for right angled isosceles triangles?

The proof is for general right triangles. An isosceles right triangle is a particular case of a general right triangle, so the proof already applies to isosceles right triangles (let $$b=a$$).

While it's true that the "waffle cone" construction requires $$a you don't need to do the construction if $$a=b.$$ The whole point of the construction is to obtain $$a^2 + b^2 = \dfrac{2ab}{\sin 2\alpha},$$ but if $$a=b$$ then $$\alpha = 45^\circ$$ and the above reduces to $$a^2 + a^2 = \dfrac{2 \cdot a \cdot a}{\sin 90^\circ}$$ which is true.

1. Can you say that approach is trigonometrical because of using trigonometrical formula $$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$$ since it also can be derived geometrically by right angled triangles?

When they call it a trigonometric proof, all they're saying is that it uses trigonometric functions. That's it.

Sure, you could write the proof without using trigonometric functions. You could reduce everything down to simple algebra on similar triangles, explicitly writing $$b/c$$ in place of $$\sin \beta.$$

But does that mean it's not trigonometric? No. You could apply the same reasoning to argue ridiculous things, e.g. that the Fundamental Theorem of Calculus

$$\int_a^b F'(x) \, \textrm dx = F(b) - F(a)$$

is not actually differential/integral calculus because you can reduce it down to just limits:

$$\lim_{n \to 0} \sum_{i=0}^{n-1} \left( \lim_{h \to 0} \frac{F(x_i +h)-F(x_i)}{h} \right) \Delta x = F(b) - F(a)$$

where $$\Delta x = \dfrac{b-a}{n}$$ and $$x_i = a + i \Delta x.$$

So, when we say the proof is trigonometric, we mean that it uses trigonometric notation to describe symbolic patterns leveraged in trigonometry, not that it is impossible to state without trigonometric notation.

1. Is it not possible to complete the proof by just adding the squares of two perpendicular sides without using the formula $$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$$ and say that proof is geometrical?
Here my question is related to how we have to update students using new developments in mathematics, therefore I hope you will not misunderstand this as a mathematics question.

This question is impossible to answer because the double-angle formula $$\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$$ is not even used in the proof.

The only trigonometry that's used is the definition of sine and the law of sines. And the law of sines does not rely on the double-angle formula -- it follows directly from the definition of sine.

• 1)Those geometric series converge if a < b only, how can you do that construction if a = b . Commented Oct 23, 2023 at 17:14
• @JanakaRodrigo you don't need to do the construction if $a=b.$ The whole point of the construction is to obtain $a^2 + b^2 = \dfrac{2ab}{\sin 2\alpha}.$ If $a=b$ then $\alpha = 45^\circ$ and $a^2 + b^2 = \dfrac{2ab}{\sin 2\alpha}$ reduces to $a^2 + a^2 = \dfrac{2 \cdot a \cdot a}{\sin 90^\circ}$ which is true. Commented Oct 23, 2023 at 17:30
• Can you use the result for $Sin2α$ to deduce the result when $a = b$ because you can get it when $a < b$ only . Commented Oct 23, 2023 at 17:59
• @JanakaRodrigo Sorry, I don't understand what you're asking. I think you might be misunderstanding the flow of the proof. Using the definition of sine, we have $c^2 = \dfrac{2ab}{\sin 2\alpha}$ regardless of the values of $a$ and $b.$ If $a=b,$ then $c^2 = \dfrac{2ab}{\sin 2\alpha} = \dfrac{2a^2}{1} = a^2 + a^2.$ (This is obvious so it's not explicitly stated in the proof.) Otherwise, if $a<b,$ then we perform the waffle cone construction to obtain $c^2 = \dfrac{2ab}{\sin 2\alpha} = a^2 + b^2.$ Commented Oct 23, 2023 at 18:12
• Yes I see , what you are saying is isosceles right angled triangles also obey the result. But you have to do it separately since it can't be directly covered by their approach. Related to another issue mentioned earlier you are correct , here you can not see double angle formula but according to the way they have obtained$Sinβ , α + β$need to be right angle and you can replace $Sinβ$ by $Cosα$and it shows how the double angle formula is used. Commented Oct 24, 2023 at 1:44