In the tables below, \(0\leq \theta \leq 2\pi\) and any missing functions are from the following list. \[\sin \theta \quad \cos \theta \quad \tan \theta \quad \sec \theta \quad \cosec \theta \quad \cot \theta\]

Some of the row and column headings are missing. Without using a calculator, try to work out what they could be and complete the table. A function does not appear twice in the same table.

If you think you know what a missing function or value is, make sure you check that it works for all the entries in its row and column!

\(\theta= \ldots\) | \(\theta= \ldots\) | \(\theta= \ldots\) | |
---|---|---|---|

\(-1\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | |

\(\tan \theta\) | undefined | \(\sqrt{3}\) | |

\(\sec \theta\) | \(-\dfrac{2}{\sqrt{3}}\) |

In the next table we have given some more information about \(\theta.\) Try to identify the missing functions and complete the table. Remember not to use a calculator!

\(\theta\) is reflex | \(\theta= \ldots\) | \(\theta\) is obtuse | |
---|---|---|---|

\(0\) | \(-\dfrac{3}{5}\) | ||

\(\cosec \theta\) | \(1\) | \(\dfrac{5}{4}\) | |

\(\dfrac{5}{12}\) | \(0\) |

How can you state the exact values of \(\theta\) in the \(1\)st and \(3\)rd columns of the second table?

How might the answers change if \(\theta\) could be any value or you could use functions like \(-\sin \theta\) in the tables?

There are some things to help you make your own ‘trig table’ in the section Design your own table.