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I am reading a paper related to astrophysics and I am stuck in a step in a calculation. It is about orbiting gas in a spiral galaxy and it calculates the errors in the fitted rotation velocity if one of the viewing angles is incorrectly estimated. My problem is that I don't understand one of the steps which has to do with either inserting a trigonometric identity or performing a Taylor expansion. The following equations are given:

[itex] x' = \hat{R} \cos(\hat{\psi})[/itex], [itex] y' = (q+\delta q) \hat{R} \sin(\hat{\psi})[/itex] as well as: [itex] \cos(\psi) =\frac{x'}{\sqrt{x'^2 + y'^2/q^2}}[/itex]

Combining these, one should arrive at the following:

[itex]\cos(\psi) = (1-\frac{\delta q}{4q}) \cos(\hat{\psi})+\frac{\delta q}{4q} \cos(3 \hat{\psi})[\itex]

I tried inserting the expressions for x' and y' into the equation for psi and then tried to apply some trigonometric identities but with no luck. In the paper it doesn't say that it is a Taylor approximation, but I can't really exclude this. I get stuck with the following expression:

[itex]\cos(\psi) = \frac{1}{\sqrt{1+(\frac{\delta q}{q} \tan(\hat{\psi}))^2}}[\itex]

Thanks very much!

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# Hi all,I am reading a paper related to astrophysics and I am stuck

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