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Reciprocal ratios

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5.4 Reciprocal ratios (EMA3Q)

Each of the three trigonometric ratios has a reciprocal. The reciprocals: cosecant (cosec), secant (\(\sec\)) and cotangent (\(\cot\)), are defined as follows:

\begin{align*} \text{cosec } \theta & = \frac{1}{\sin\theta } \\ \sec\theta & = \frac{1}{\cos\theta } \\ \cot \theta & = \frac{1}{\tan\theta } \end{align*}

We can also define these reciprocals for any right-angled triangle:

\begin{align*} \text{cosec } \theta & = \frac{\text{hypotenuse}}{\text{opposite}} \\ \sec \theta & = \frac{\text{hypotenuse}}{\text{adjacent}} \\ \cot \theta & = \frac{\text{adjacent}}{\text{opposite}} \end{align*}

Note that:

\begin{align*} \sin \theta \times \text{cosec } \theta & = 1\\ \cos \theta \times \sec \theta & = 1\\ \tan \theta \times \cot \theta & = 1 \end{align*}

This video covers the three reciprocal ratios for \(\sin\), \(\cos\) and \(\tan\).

Video: 2FNV

You may see cosecant abbreviated as \(\csc\).