I don't understand where does this equality come from: $$\arctan \Bigl (\frac {\sin x} {\cos x +3}\Bigl)= \mathrm {Im} \log (i\sin x + \cos x +3).$$ I guess it's linked to the more famous equality relating $\arctan $ and complex logarithm, however I still don't see the point. Thanks for any explanation
2026-05-14 14:59:13.1778770753
Complex equality
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Complex logarithm is not uniquely defined so the statement is not correct. However one value of the imaginary part of $log (a+ib)$ is $\arctan (\frac b a)$ so one value of RHS equals LHS in your equation.