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The equation $$\DeclareMathOperator{\Dd}{d\!} \frac{\Dd y}{\Dd x} = \tan (\theta \pm \mu)\label{1}\tag{1} $$ is not derived explicitly in the book "Introduction to Numerical Techniques for Nonlinear Supersonic Flow" by C. K. Chu. The book focuses on numerical techniques for solving nonlinear supersonic flow equations rather than providing a derivation for the equation.

As a result, I would like to derive the trigonometric equation \eqref{1}, as prescribed in the book.

  • I attached an image here from the textbook for your reference. Eq. 13.10 is the equation, which I would like to derive from Eq. 13.9.
  • I'm hereby attaching what I have done. I wasn't able to make it to the final possible result.

Can someone guide/correct me? Thank you

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  • $\begingroup$ Hello Gonçalo, Thanks for your response. I attached the textbook reference for your understanding. Kindly check it $\endgroup$
    – Kumaresh
    Commented Jul 5, 2023 at 4:58

2 Answers 2

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Your mistake is writing that $$\sin 2\theta + \sin 2 \mu \ne \sin 2(\theta + \mu).$$ For example, with $\theta = \mu = \frac \pi 4$, \begin{align*} \sin 2 \theta + \sin 2 \mu &= \sin \frac \pi 2 + \sin \frac \pi 2 = 2, \text{ but}\\ \sin2(\theta + \mu) &= \sin \pi = 0. \end{align*} You got as far as $$\frac{- \sin 2 \theta \pm \sin 2 \mu}{\cos 2 \theta + \cos 2 \mu},$$ which is correct.

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  • $\begingroup$ Thank you kipf. Got my solution when you pointed out my mistake. Attaching the solution in the form of image here. $\endgroup$
    – Kumaresh
    Commented Jul 5, 2023 at 10:49
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The solution to this query. Thank you Soution

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  • $\begingroup$ Welcome to MSE. It is in your best interest that you type your posts (using MathJax) instead of posting links to pictures. $\endgroup$ Commented Jul 5, 2023 at 10:58
  • $\begingroup$ Thank you. I'll do that from my next post. :-) $\endgroup$
    – Kumaresh
    Commented Jul 6, 2023 at 1:26

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