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This question is a follow up question to this answer.

In the equation:

$$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta.$$

$a$ and $b$ are given. What is the best way to solve for $\theta$? If a direct solution is not possible which numerical method do you suggest?

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up vote 2 down vote accepted

From $$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta$$ we get $$a\cos^2\theta = \frac{b}{2}\cot\theta(2\cos^2\theta-1) = \frac{b}{2}\cot\theta\cos 2\theta $$ With $\cot\theta = \frac{\cos\theta}{\sin\theta}$ and $\cos\theta\neq 0$ this gives

$$a\sin\theta\cos\theta = \frac{b}{2}\cos 2\theta$$ which finally results in $$\frac{a}{2}\sin 2\theta = \frac{b}{2}\cos 2\theta$$ and consequently $$\tan 2\theta = \frac{b}{a}$$

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