One day a long time ago I stumbled across the law of tangents which states that $\frac{\tan{\frac{A-B}{2}}}{\tan\frac{A+B}{2}}=\frac{a-b}{a+b}$ now as my question states what possible use could I have for this? I was thinking about it and anything I could think about using the law of tangents for could just be done using the law of sines much faster and easier. And here's the kicker, the LHS can be reduced down to an alternative law of sines $\frac{\sin A-\sin B}{\sin A+\sin B}$ so how could the law of tangents be useful? Perhaps in a proof? No disrespect to the guy who came up with it of course, it is an interesting concept.
Finding a use for the law of tangents
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algebra-precalculus
trigonometry
soft-question
1 Answers
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These trig laws are mostly for solving triangles. (Finding unknown angles and sides from known ones.) If you're in a $SAS$ case, then the law of cosines is usually called for. However, if the two sides are almost equal, the subtraction in the law of cosines plus the extraction of the square root means a unwieldy loss of significant digits. And there's that square root. So computationally, law of tangents is much more efficient and more accurate. Use
$$\tan \left(\frac{A-B}{2}\right) = \frac{a-b}{a+b}\cot\left( \frac{C}{2} \right)$$
to find $C$.