Given the following trigonometric formulae, use Osborn's rule to write down the corresponding hyperbolic function formulae.
$sin 3 A = 3 sin A - 4 {sin}^{3} A$

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Solution

Osborn's rule is a rule for converting a trigonometric identity into corresponding hyperbolic one.The rule states that one replaces every occurence of sine or cosine with the corresponding hyperbolic sine or cosine and whenever one has a product of two sines ,the product of hyperbolic sines must be negated.
Thus ;
$s i n 3 A = 3 s i n A - 4 s i n^{3} A$

$s i n^{A} \to - s i n h^{2} A$

Hence;
Corresponding Hyperbolic identity:

$s i n 3 A = 3 s i n h A + 4 s i n h^{3} A$

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