In the ambiguous case, given a, b, and A, prove that the difference between the two values of c is $2 \sqrt{a^{2} - b^{2} s i n^{2} A} .$

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Solution

We have $a^{2} = b^{2} + c^{2} - 2 b c c o s A$
or $c^{2} - 2 b c c o s A + b^{2} - a^{2} = 0$
Let the two values of c be $c_{1}$ and $c_{2}$ . Then $c_{1} + c_{2} = 2 b c o s A$ and $c_{1} c_{2} = b^{2} - a^{2} .$
Hence $(c_{1} - c_{2})^{2} = (c_{1} + c_{2})^{2} - 4 c_{1} c_{2}$
= $4 b^{2} c o s^{2} A - 4 (b^{2} - a^{2})$
= $4 [a^{2} - b^{2} (1 - c o s^{2} A)]$
= $4 [a^{2} - b^{2} s i n^{2} A]$
$∴ c_{1} - c_{2} = 2 \sqrt{a^{2} - b^{2} s i n^{2} A}$

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