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In teaching the high school curriculum Pythagoras is usually bundled with Trigonometry. They might be justified by way of proof of some sort. They are used to solve 2d and 3d geometry problems for unknowns.

They are bundled together with the justification that they both are used for funding unknowns in right triangles. I would argue against this pairing on the basis that their similarities are superficial and their differences are more fundamental.

In fact, either could be taught first with their own respective prerequisites.

What arguments are some arguments for their sequencing?

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    $\begingroup$ The premise that these are "usually bundled" seems unfounded. E.g., Common Core standards for the 8th grade cover the Pythagorean Theorem, but there's no reference to any trigonometry. $\endgroup$ Commented Jul 6, 2023 at 16:43
  • $\begingroup$ Right triangle and Pythagoras theorem is considered a part of geometry where I am from, I think it is 8th grade. The notion of sine, cosine and tangent are introduced, but little more. Trigonometry is considered a more complicated topic and is part of algebra, and is studied a couple of years later. $\endgroup$
    – Rusty Core
    Commented Jul 7, 2023 at 18:16
  • $\begingroup$ When do you cover similar triangles? $\endgroup$ Commented Jul 8, 2023 at 15:41

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I learnt about Pythagoras before I started learning about trigonometry.
I have learnt the Pythagoras proof, using uniform triangles, but there is also this very popular proof of the Pythagoras theorem:

enter image description here

The whole idea is based on the total surface, being equal to the sum of the individual surfaces:

$$S(square(A+B)) = S(square(C)) + 4 \cdot S(triangle(A,B))$$
$$(A+B)^2 = C^2 + 4 \cdot \frac{A \cdot B}{2}$$
$$A^2 + 2 \cdot A \cdot B + B^2 = C^2 + 2 \cdot A \cdot B$$

Hence:

$$A^2 + B^2 = C^2$$

As you see, no $\sin$, $\cos$ or $\tan$ involved :-)

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    $\begingroup$ This doesn't seem to be an answer to the question which was asked. What does this have to do with the sequencing of the Pythagorean theorem and trigonometry? $\endgroup$
    – Xander Henderson
    Commented Jul 6, 2023 at 14:15
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    $\begingroup$ @XanderHenderson: the question is about teaching both subjects at the same time or separately. My answer proves that you don't need trigonometry in order to talk about Pythagoras. $\endgroup$
    – Dominique
    Commented Jul 7, 2023 at 6:54
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    $\begingroup$ My answer proves that you don't need trigonometry in order to talk about Pythagoras. --- Since the OP already acknowledged this by saying "In fact, either could be taught first with their own respective prerequisites.", I don't see how your comment addresses the issue that Xander Henderson raised. $\endgroup$ Commented Jul 7, 2023 at 11:27
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    $\begingroup$ The author has made a statement (s)he wishes to spread. In order to do so, (s)he needs authority. My answer proves his or her right and as such increases that authority. $\endgroup$
    – Dominique
    Commented Jul 7, 2023 at 11:51
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    $\begingroup$ I saw this video youtube.com/watch?v=8jJONUXGsNo and I wondered what was some figures showed in the video when teaching Pythagoras theorem, and in this answer I found the answer. Thank you @Dominique $\endgroup$ Commented Apr 14 at 17:48

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