Have a sine

Have a sine

Excerpt: Students express several lengths in a unit circle diagram using trigonometric functions and thereby develop their understanding of how trigonometric functions are related to each other. This problem is accessible to any student with a basic understanding of trigonometry in right-angled triangles and offers opportunities to discuss what happens if $\theta$ varies. There is also a historical link with names for expressions such as $\tfrac{1}{2}(1-\cos \theta).$

Problem

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 $\cos \theta$.

1. $OB$

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1. $AB$

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1. $BC$ (this was known as the versed sine or versine of $\theta$, written $\mathop{\mathrm{versin}}\theta$)

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1. $OD$

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1. $CD$ (this was known as the exsecant of $\theta$, written $\mathop{\mathrm{exsec}}\theta$)

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1. $AD$

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1. $AE$

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1. $AG$

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1. $GO$

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1. $FG$ (this was known as the coversed sine or coversine of $\theta$, written $\mathop{\mathrm{coversin}}\theta$ or $\mathop{\mathrm{covers}}\theta$ or $\mathop{\mathrm{cvs}}\theta$)

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1. $EF$ (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?

1. The versed cosine or vercosine of $\theta$, written as $\mathop{\mathrm{vercos}}\theta$, which is defined by $\mathop{\mathrm{vercos}}\theta = 1 + \cos\theta$

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1. 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 $\cos\theta = \sin\left(\frac{\pi}{2} - \theta\right)$.

1. How are versine and coversine related?

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1. How are vercosine and covercosine related?

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

Suggestion

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.

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

We have one angle $\theta$ given on the diagram, so which lengths can we find using that?

Are there any other angles that are also $\theta$?

How many right-angled triangles are there in the diagram?

Solution

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.

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

The triangle $OAB$ is right-angled, with a right angle at $B$ and with hypotenuse of length $1$, so we see that $OB = \cos\theta.$

Similarly, we have $AB = \sin\theta.$

Since the circle is a unit circle, we know that $OC = 1$ and so $BC = OC - OB = 1 - \cos \theta.$ This tells us that $\mathop{\mathrm{versin}}\theta = 1 - \cos \theta.$

The triangle $OAD$ is right-angled, with a right angle at $A$ and with $\angle DOA = \theta$, and the side $OA$ has length $1$, so we find that $\cos\theta = \dfrac{1}{OD},$ so $OD = \dfrac{1}{\cos\theta}.$

Therefore we have $CD = OD-OC = \dfrac{1}{\cos\theta} - 1 = \sec\theta - 1.$ This means we can write $\mathop{\mathrm{exsec}}\theta = \sec\theta - 1.$

Looking again at triangle $OAD$, we find that $AD = \tan\theta = \dfrac{\sin\theta}{\cos\theta}.$

We can consider triangle $AOE$, which has a right angle at $A$ and which has angle $\theta$ at $E$. The side $AO$ has length $1$, and so we see that $\tan\theta = \dfrac{1}{AE},$ and therefore $AE = \cot\theta = \dfrac{\cos\theta}{\sin\theta}.$

Triangle $AGE$ has a right angle at $G$ and angle $\theta$ at $E$, and we have just seen that its hypotenuse has length $\cot\theta$. So $AG = \cot\theta \sin\theta = \cos\theta.$

Quadrilateral $ABOG$ is a rectangle, because its angles are all right angles, so $GO = AB = \sin\theta.$

The circle has radius $1$, so $FG = FO - GO = 1 - \sin\theta.$ Therefore $\mathop{\mathrm{coversin}}\theta = 1 - \sin\theta.$

We can find the length $EO$ using the right-angled triangle $DOE$, which has an angle $\theta$ (at $E$). We know that $OD = \sec\theta$, and so we have $EO = \dfrac{\sec\theta}{\tan\theta} = \dfrac{1}{\sin\theta}.$

Then we have $EF = EO - 1 = \dfrac{1}{\sin\theta} - 1.$ Therefore we can write $\mathop{\mathrm{excosec}}\theta= \cos ec \theta -1.$

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

1. The versed cosine or vercosine of $\theta$, written as $\mathop{\mathrm{vercos}}\theta$, which is defined by $\mathop{\mathrm{vercos}}\theta = 1 + \cos\theta.$

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1. The coversed cosine or covercosine of $\theta$, written as $\mathop{\mathrm{covercos}}\theta$, which is defined by $\mathop{\mathrm{covercos}}\theta = 1 + \sin\theta.$

Since the circle has radius $1$ and $OB = \cos\theta$, the line segment $BH$ has length $1 + \cos\theta$.

Similarly, $GJ$ has length $1 + \sin\theta$.

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

1. How are versine and coversine related?

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1. How are vercosine and covercosine related?

We have

$$\begin{align*} \mathop{\mathrm{coversin}}\left(\dfrac{\pi}{2} - \theta\right) &= 1 - \sin\left(\dfrac{\pi}{2} - \theta\right) \\ &= 1 - \cos\theta \\ &= \mathop{\mathrm{versin}}\theta. \end{align*}$$

Similarly, we have

$$\begin{align*} \mathop{\mathrm{covercos}}\left(\dfrac{\pi}{2} - \theta\right) &= 1 + \sin\left(\dfrac{\pi}{2} - \theta\right) \\ &= 1 + \cos\theta \\ &= \mathop{\mathrm{vercos}}\theta. \end{align*}$$

The prefix “co-” probably originates from the word “complementary”. Two angles are called complementary if they add up to a right angle. So:

• the cosine of an angle is the sine of the complementary angle

• the cosecant (cosec) of an angle is the secant (sec) of the complementary angle

• the cotangent (cotan) of an angle is the tangent (tan) of the complementary angle

• the coversine of an angle is the versine of the complementary angle

• the covercosine of an angle is the vercosine of the complementary angle