Fluency exercise

## Problem

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?