Home Practice
For learners and parents For teachers and schools
Textbooks
Full catalogue
Leaderboards
Learners Leaderboard Classes/Grades Leaderboard Schools Leaderboard
Pricing Support
Help centre Contact us
Log in

We think you are located in United States. Is this correct?

5.4 Reciprocal ratios

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 \(\csc\).

temp text