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.