MHB An application of the law of cosines?

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The discussion revolves around the application of the law of cosines in a mathematical problem related to music theory and Plato's works. The original poster, a philologist with basic math knowledge, seeks clarification on a formula presented by Dom Néroman. The analysis reveals that the formula contains an ambiguous $\pm$ sign, which differentiates between two ratios, and identifies a potential error regarding the inclusion of $PH$ in the formula. The corrected expressions for the ratios are provided, emphasizing the need for precision in mathematical representation. Overall, the conversation highlights the intersection of mathematics and music theory while addressing specific mathematical inaccuracies.
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Hello, everybody:

I am a philologist who is fond of mathematics, but who unfortunately has just an elementary high school knowledge of them. I am translating La leçon de Platon, by Dom Néroman (La Bégude de Mazenc, Arma Artis, 2002), which deals with music theory and mathematics in the works of Plato. The problem which brings me here is not about translation, but about mathematics. I am attaching a file with my problem and what I have tried so far to solve it.

Thank you very much in advance and best greetings.
 

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In fact, your answer is essentially the same as Néroman's. You have shown that $$\begin{aligned}\frac{PM}{PH} &= \frac{6\pm\sqrt{36 + 64(1 + \tan^2i)}}{8\sqrt{1+\tan^2i}} \\ &= \frac{6\pm\sqrt{100 + 64\tan^2i)}}{8\sqrt{1+\tan^2i}} \\ &= \frac{6\pm\sqrt{4(25 + 16\tan^2i)}}{8\sqrt{1+\tan^2i}} \\ &= \frac{6\pm\sqrt{4}\sqrt{25 + 16\tan^2i}}{8\sqrt{1+\tan^2i}} \\ &= \frac{6\pm2\sqrt{25 + 16\tan^2i}}{8\sqrt{1+\tan^2i}}. \end{aligned}$$ Now divide top and bottom of the fraction by $2$ to get $$\frac{PM}{PH} = \frac{3\pm\sqrt{25 + 16\tan^2i}}{4\sqrt{1+\tan^2i}}.$$ That is exactly Néroman's formula apart from two things. One of these is the ambiguous $\pm$ sign. That is resolved by the fact that the $+$ sign gives the ratio $\frac{PM}{PH}$, and the $-$ sign gives the ratio $\frac{Pm}{PH}$. The other anomaly is the $PH$ (or $h$) that occurs in Néroman's formula. That must be a mistake on his part. The formula should be either $$\tfrac{PM}{PH} = \tfrac1{4\sqrt{1+\tan^2i}}\bigl(3 + \sqrt{25 + 16\tan^2i}\bigr)$$ or $$PM = \tfrac{PH}{4\sqrt{1+\tan^2i}}\bigl(3 + \sqrt{25 + 16\tan^2i}\bigr).$$ But he appears to have mistakenly put the $PH$ on both sides of the formula.
 
Opalg said:
In fact, your answer is essentially the same as Néroman's. You have shown that $$\begin{aligned}\frac{PM}{PH} &= \frac{6\pm\sqrt{36 + 64(1 + \tan^2i)}}{8\sqrt{1+\tan^2i}} \\ &= \frac{6\pm\sqrt{100 + 64\tan^2i)}}{8\sqrt{1+\tan^2i}} \\ &= \frac{6\pm\sqrt{4(25 + 16\tan^2i)}}{8\sqrt{1+\tan^2i}} \\ &= \frac{6\pm\sqrt{4}\sqrt{25 + 16\tan^2i}}{8\sqrt{1+\tan^2i}} \\ &= \frac{6\pm2\sqrt{25 + 16\tan^2i}}{8\sqrt{1+\tan^2i}}. \end{aligned}$$ Now divide top and bottom of the fraction by $2$ to get $$\frac{PM}{PH} = \frac{3\pm\sqrt{25 + 16\tan^2i}}{4\sqrt{1+\tan^2i}}.$$ That is exactly Néroman's formula apart from two things. One of these is the ambiguous $\pm$ sign. That is resolved by the fact that the $+$ sign gives the ratio $\frac{PM}{PH}$, and the $-$ sign gives the ratio $\frac{Pm}{PH}$. The other anomaly is the $PH$ (or $h$) that occurs in Néroman's formula. That must be a mistake on his part. The formula should be either $$\tfrac{PM}{PH} = \tfrac1{4\sqrt{1+\tan^2i}}\bigl(3 + \sqrt{25 + 16\tan^2i}\bigr)$$ or $$PM = \tfrac{PH}{4\sqrt{1+\tan^2i}}\bigl(3 + \sqrt{25 + 16\tan^2i}\bigr).$$ But he appears to have mistakenly put the $PH$ on both sides of the formula.
Thank you very much!
 
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