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March 22[edit]

Sin, cos and ellipses[edit]

In this book [1] (linked to the right page), left column towards the bottom. I'm having a problem with the "hence."

I understand that:

A: the point M has the coordinates (x,y), which is also (sin φ, cos φ), no matter how you slide the straight KL, for all φ.

B: the formula for the ellipse.

{\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.

I just don't get how B follows from A. Maybe I'm missing a concept that the authors take for granted. But shouldn't the text have explained something in between? Something like why the ellipse matches the Pythagorean trigonometric identity? Or why the set of all possibles values of M is $\sin ^{2}\theta +\cos ^{2}\theta =1$ ? Why is that when you have the coordinates (sin, cos) you add them and equal to 1 to produce an ellipse? Grapesofmath (talk) 17:27, 22 March 2024 (UTC)[reply]

I think you've misread the text: the point M (x, y) is actually (a sin φ, b cos φ). So x/a = sin φ and y/b = cos φ. Substituting into the identity

\sin ^{2}\phi +\cos ^{2}\phi =1

(which is true for all φ) gives the formula in (B). AndrewWTaylor (talk) 18:09, 22 March 2024 (UTC)[reply]

More precisely, the text identifies the point M (x; y) by "y = b sin φ and x = a cos φ". The resulting equation is the same. --Lambiam 19:32, 22 March 2024 (UTC)[reply]

Thanks for the feedback. But shouldn't the text be explicit here and explain that A applied to Pythagorean trigonometric identity result in B? Isn't this a jump too big in the train of thought? Grapesofmath (talk) 23:59, 22 March 2024 (UTC)[reply]

It's not immediately obvious, but it's not that hard either given the diagram. The y = b sin φ come from the lower right triangle and x = a cos φ comes from the upper left triangle. Once you have those equations, the equation (1) in the text follows as explained above. I think as a reader you're meant to figure out this kind of detail yourself. The alternative would be that the text becomes long-winded and pedantic. It's also a better learning experience for the reader if they have to think about the text as they're reading it and fill in some missing steps. A lot depends on the intended audience as well; apparently the book is meant for people with a certain basic knowledge of geometry, perhaps with some experience writing proofs. (That kind of information is often given in an introduction, but this is the introduction.) --RDBury (talk) 04:17, 23 March 2024 (UTC)[reply]