The trigonometric functions with which we are most familiar are sine and cosine. From them, we can obtain tangent, and also the reciprocals secant, cosecant and cotangent. In the past, mathematicians have used many other trigonometric functions too. Some of these are illustrated below. (You can read more about the history and use of these in navigation in Lost but lovely: the haversine.)

Find the following lengths, in terms of \(\sin\theta\) and \(\cos \theta\).

\(OB\)

\(AB\)

\(BC\) (this was known as the

*versed sine*or*versine*of \(\theta\), written \(\mathop{\mathrm{versin}}\theta\))\(OD\)

\(CD\) (this was known as the

*exsecant*of \(\theta\), written \(\mathop{\mathrm{exsec}}\theta\))\(AD\)

\(AE\)

\(AG\)

\(GO\)

\(FG\) (this was known as the

*coversed sine*or*coversine*of \(\theta\), written \(\mathop{\mathrm{coversin}}\theta\) or \(\mathop{\mathrm{covers}}\theta\) or \(\mathop{\mathrm{cvs}}\theta\))\(EF\) (this was known as the

*excosecant*of \(\theta\), written \(\mathop{\mathrm{excosec}}\theta\) or \(\mathop{\mathrm{excsc}}\theta\))

Can you find line segments on the diagram that have the following lengths?

The

*versed cosine*or*vercosine*of \(\theta\), written as \(\mathop{\mathrm{vercos}}\theta\), which is defined by \[\mathop{\mathrm{vercos}}\theta = 1 + \cos\theta\]The

*coversed cosine*or*covercosine*of \(\theta\), written as \(\mathop{\mathrm{covercos}}\theta\), which is defined by \[\mathop{\mathrm{covercos}}\theta = 1 + \sin\theta\]

We know that \(\cos\theta = \sin\left(\frac{\pi}{2} - \theta\right)\).

How are versine and coversine related?

How are vercosine and covercosine related?

What do you think that the prefix “co-” might indicate here?