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Reciprocal trigonometric ratios are ratios that involve the reciprocal (or multiplicative inverse) of the trigonometric functions. The reciprocal ratios are the cosecant (cosec or csc), secant (sec), and cotangent (cot).
The formulas for the reciprocal ratios are as follows:
$$ \csc\theta = \frac{1}{\sin\theta} \\ \sec\theta = \frac{1}{\cos\theta} \\ \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} $$
Here is an example:
Given that \( \sin\theta = \frac{3}{5} \), find the values of \( \csc\theta \), \(\sec\theta \), and \(\cot\theta\).
$$ \csc\theta = \frac{1}{\sin\theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3} $$ $$ \sec\theta = \frac{1}{\cos\theta} = \frac{1}{\sqrt{1 - \sin^2\theta}} = \frac{1}{\sqrt{1 - \left(\frac{3}{5}\right)^2}} = \frac{5}{4} $$ $$ \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{\sqrt{1 - \sin^2\theta}}{\sin\theta} $$ $$= \frac{\sqrt{1 - \left(\frac{3}{5}\right)^2}}{\frac{3}{5}} = \frac{4}{3} $$Therefore, \( \csc\theta = \frac{5}{3}\), \(\sec\theta = \frac{5}{4}\), and \(\cot\theta = \frac{4}{3}\).
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