Complex Numbers Birth Trigonometry!

Measuring Angles of Rotation in a Dimensionless Way

Recall that we have interpreted complex multiplication in the context of rotating points in a plane representing complex numbers about the origin. Such rotations may be some part of a full rotation, a full rotation or more, or even rotations in the opposite direction (when the associated angles are negative). Exactly how much we rotate has so far been described in terms of degrees.

Hipparchus of Rhodes
Most will be familiar with measuring angles in degrees, but what many may not know is that this unit of measurement traces back to the second century BC, when the Greek astronomer and mathematician Hipparchos of Rhodes began applying geometry to Babylonian astronomy (likely due to the amazing collection of astronomical data the Babylonians had compiled -- i.e., 800 years worth of nightly written records).

Despite the existence at the time of Euclid's Elements (c. 300 BC), a text which largely set the standard for learning geometry -- that work never provided a unit of measurement for angles besides the right angle. Hipparchos thus borrowed the Babylonian division of the ecliptic -- a great circle on the celestial sphere representing the sun's apparent path during the year, so called because lunar and solar eclipses can occur only when the moon crosses it.

The ancient Babylonians had divided this circular path into 12 sections called beru (interestingly, with names that in Greek translate to Gemini, Cancer, Leo, etc.). Then, they divided each of these sections into 30 equal subsections -- allowing the position of the sun to be described in one of $12 \cdot 30 = 360$ ways. Thus, the notion of a degree -- one $360^{th}$ of a full rotation -- was born. Why $12$ sections and $30$ subdivision you ask? Likely this was due to the fact that the calendar Babylonians used at the time was based on lunar cycles (cycles of the moon). Each such cycle lasted approximately 30 days (i.e., sometimes 29, othertimes 30), with $12$ lunar cycles occurring with each year.

All that fascinating history aside, there are better things we can use to compare the sizes of angles than the number of days in a lunar cycle!

As a starting place, consider the length of the path that some complex $z$ creates as it moves from $1$ counter-clockwise around the unit circle. Recall that one complete rotation about $0$ (the origin) should result in a path $2\pi$ times as long as the radius. But how long is that, actually -- are we measuring in inches? feet?

Rather than try to pick some arbitrary dimension for the measure for these lengths (which is actually not unlike the somewhat arbitrary Babylonian division of a full rotation into 360 degrees), what if we instead define the angle measure by the ratio of the length of the path $z$ takes, and the length of the radius? Note that as both would presumably be measured in the same dimension, allowing these to cancel -- leaving a "dimensionless" radian measure of the angle in question.

For clarity (especially when using both degrees and radians), we sometimes write angles measures expressed in radians as $\theta$ rad, but given that radians are actually dimensionless, we will more often omit the "rad" part.

Measuring angles in this way equates a full rotation of $360^{\circ}$ to a radian measure of $2\pi$, and all other angles to their proportional equivalents (for example, half of $360^{\circ}$ is $180^{\circ}$ and half of $2\pi$ is $\pi$, so $180^{\circ}$ and $\pi$ are equal angle measures. Rotations of some common radian measures are shown below:

Given that a radian measure of $\pi$ is the same as $180^{\circ}$ and dividing both sides by $180$, we quickly see that $1^{\circ} = \frac{\pi}{180}$, we can then use this to easily convert the degree measure of any angle to radians and the radian measure of any angle to degrees. Some example conversions are shown below:

Precisely because they do not rely on any (arbitrarily) chosen dimension, the use of radians to measure angles will greatly simplify many things in both calculus and other areas of mathematics.

The Cosine and Sine Functions

We saw when we introduced complex numbers previously that multiplication of one complex number by another involves both rotation about the origin and scaling the distance from the same. Of course, if the complex numbers involved were both of unit magnitude, then only rotation was involved.

With this in mind, let us consider again the unit circle of complex values with unit magnitude. In particular, let us consider the real and imaginary parts of the $z$ on that unit circle where $\theta = arg(z)$ is known. We should recognize that knowing $\theta$ fixes exactly where this $z$ must be located in the complex plane. To see this, imagine one starts at $0$ (the origin) and then moves exactly one unit distance in a direction corresponding to $\theta$.

The real part of such a $z$ and its imaginary coefficient (and consequently the $x$ and $y$ coordinates of the corresponding point in the related coordinate plane -- also called the Cartesian plane, named after French mathematician Rene Descartes.) are also then fixed once we know $\theta$. In this way, we can think of these $x$ and $y$ coordinates as functions of $\theta$. To attach specific names to these functions, let us make the following definitions:

For any $\theta$, let $z = x + iy$ be the unique complex value on the unit circle (i.e., $|z| = 1$) with argument $\theta$. Then, define functions cosine and sine, denoted* $\cos \theta$ and $\sin \theta $ respectively, so that: $$\cos \theta = x = Re(z) \quad \textrm{ and } \quad \sin \theta = y = Im(z)$$

*Note that -- just like log functions -- we traditionally omit the parentheses around the input when doing so doesn't cause confusion.

To visualize what these functions do, consider the diagram drawn on the left below. Of course, we can also (and more frequently do) draw such things instead on the related coordinate plane, as shown on the below right.


You may be curious about why we have included the shaded blue triangle in the diagram. This triangle -- formed by drawing a vertical line from the point on the unit circle to the real axis (or $x$-axis on the right) is called a reference triangle, and serves as a way to quickly compute sine and cosine values for certain commonly encountered angles. We will have more to say about it shortly.

For now, let us simply say that the cosine and sine functions form the basis of the area of mathematics called trigonometry, a word which traces back to the Greek words trigonan which means "triangle" and metron, meaning "to measure". In many ways, this reference triangle can be thought of as the triangle to which the word "trigonometry" owes its origin!

Some Basic Properties of the Sine and Cosine Functions

Whenever we develop a new function, there are always questions we should ask -- things like what's the domain? ..what's the image/range? ..what interesting properties does it have? ..etc. The below address some of these. Given the above equivalent ways to interpret the cosine and sine functions -- as related to the complex plane or the Cartesian plane -- let us opt for the latter in what we have to say below. That is to say, let us interpret $\cos(\theta)$ for the moment as an $x$-coordinate and $\sin \theta$ as a $y$-coordinate:

Exploiting Familiar Triangles and Symmetries

For some commonly-encountered angles of $\theta$, the values of $\cos \theta$ and $\sin \theta$ are easily found using geometry. Using these values in conjunction with the symmetries seen in the unit circle, the values corresponding to other commonly encountered angles can also be easily found. We explore this idea below:

On Names and (Other) Trigonometric Functions

One might naturally wonder why the functions discussed above are named sine and cosine. The answer is interesting, in that it provides an example of how some mathematical ideas that originated in India were picked up first by the Muslims and then finally spread to Europe. The origin of sine traces back to the Sanscrit word jiya, which means "bowstring".

It is not hard to see why -- consider the images below. If the circle on the right is a unit circle, note how the sine value associated with $\theta = m\angle BOC$ is the half the length of the (undrawn) "bowstring", segment $BC$. Coincidently, in the same image we can even see the "bow" as arc $\stackrel{\mbox{$\frown$}}{BAC}$, and the arrow $OA$ notched at $O$ to the (drawn) bowstring formed by the union of radii $OB$ and $OC$!


In Arabic, however the bowstring is called jiba, although vowels are not always written in Arabic, and thus someone like $12^{th}$-century Gherardo of Cremona, who was translating an Arabic text on geometry, would have seen simply the Arabic equivalent to the letters "jb". As a consequence, Gherado failed to translate this word correctly, thinking it was another (i.e., jaib) -- a word that means "curve, fold, or hollow". In Latin, the equivalent for "curve" is sinus, and from there one can more easily see the final evolution into sine upon one more translation into English.

With regard to the origins of the word cosine, consider the following diagrams:

Clearly, $\theta_1$ and $\theta_2$ are complementary angles since their measures add to $90^{\circ}$ (or equivalently, their radian measures add to $\frac{\pi}{2}$).

Note that the two triangles containing angles marked in blue must be congruent as they are reflections of each other across the line $y=x$. The same can be said of the triangles containing angles marked in red. As such, we can see there is a relation between the sine and cosine of an angle $\theta_1$ and its complement $\theta_2$. Namely, $$\cos \theta_1 = \sin \theta_2 \quad \textrm{ and } \quad \sin \theta_1 = \cos \theta_2$$ Given complement to any angle $\theta$ (in radians) is $\frac{\pi}{2} - \theta$, we can equivalently say for any $\theta$ that

$$\textstyle{\cos \theta = \sin (\frac{\pi}{2} - \theta) \quad \textrm{ and } \quad \sin \theta = \cos (\frac{\pi}{2} - \theta)}$$

In this way, we see that the cosine of an angle is the sine of its complement. This important relationship is actually captured in the name of the cosine function. In Medieval Latin this function was expressed as complementi sinus (note the use of the word sinus discussed earlier). Around 1620, English mathematician Edmund Gunter abbreviated this with co.sinus -- which ultimately was contracted to the "cosine" we use today.

Recall, we defined the cosine and sine functions of an angle $\theta$ as the $x$ and $y$ coordinates of the point $(a,b)$ on the unit circle with $\arg(a+b) = \theta$. However, this is not the only route we could have taken to relate $\theta$, a point on the unit circle, and a unique pair of $x$ and $y$ values.

Rather than focusing on the coordinates of the point itself, perhaps we draw instead a tangent to the point in question and notice the $x$ and $y$ values where this tangent cuts through the two axes, as seen in the picture below at the points labeled $x$ and $y$.

We call the function that gives the $x$ coordinate where this tangent cuts through the $x$-axis the secant function, denoted by $\sec \theta$, noting that the Latin word secare means "to cut" into pieces, not unlike the modern word "section" (when used as a verb).

In a related way, we call the function that gives the $y$-coordinate where this tangent cuts through the $y$ axis the cosecant function, denoting this by $\csc \theta$, given that the relationship between the secant of an angle and its complement is similar to that of the sine of an angle and its complement. To see this, again note that under a reflection of the entire image above over the line $y=x$, the terminal side of the angle $\theta$ (i.e., the one labeled with unit length) would move to the same for the complement of $\theta$, and the red segment would fall on the $x$-axis, as the blue segment is now.

$$\textstyle{\sec \theta = \csc (\frac{\pi}{2} - \theta) \quad \textrm{ and } \quad \csc \theta = \sec (\frac{\pi}{2} - \theta)}$$

Alternatively, we can see this relation between complements upon discovering the relationship between the secant and cosine functions. Notice that one can easily argue $\triangle BZO$ above is similar to $\triangle ZPO$, upon which we have the proportion $\sec \theta = \frac{1}{c}$.

Recognizing the convenience of picking an angle in the first quadrant, which keeps $c = \cos \theta$ and $s = \sin \theta$ (note $c$ and $s$ are distances and thus always positive, but $\cos \theta$ and $\sin \theta$ are coordinates which can sometimes be negative.), we see that in this context the secant is simply the reciprocal of the cosine function.

In the same manner, using the similarity of $\triangle OZR$ and $\triangle ZPO$, we can argue $\csc \theta = \frac{1}{s}$. Nicely, the argument above easily extends to other quadrants, making this relationship true in general:

$$\sec \theta = \frac{1}{\cos \theta} \quad \textrm{ and } \quad \csc \theta = \frac{1}{\sin \theta}$$

Of course, drawing the tangent segment $\overline{RB}$ to the point $Z$ makes us wonder something. How long are the two segments $\overline{ZR}$ and $\overline{ZB}$ (shown below in orange and magenta, respectively) that comprise it?

Let us focus on $\overline{ZB}$ first. Note that $\triangle BZO$ must be similar to $\triangle ZPO$, so $ZB = \frac{s}{c}$, suggesting (at least for this quadrant) that $ZB$ is the quotient of the related sine and cosine values.

In other quadrants, we note that the quotient of the sine and cosine values can sometimes be negative (as they are both coordinates), but the segments produced there always have positive distance. Still, in these other quadrants similar arguments establish that the magnitudes of these two things always agree.

Using a similar argument involving $\triangle RZO$ and $\triangle ZPO$ establishes the length $ZR$ and the magnitude of $\cos \theta$ divided by $\sin \theta$ (i.e., the reciprocal of $ZB$).

As the lengths we are finding are both on the tangent line to the unit circle at the point corresponding to $\theta$, let us call the functions of $\theta$ that produce them the tangent and cotangent functions, denoted $\tan \theta$ and $\cot \theta$, defining these in the following way:

$$\tan \theta = \frac{\sin \theta}{\cos \theta} \quad \textrm{ and } \quad \cot \theta = \frac{\cos \theta}{\sin \theta}$$

The typical "co-function" relationship seen for other trigonometric function pairs (i.e., sine and cosine, secant and cosecant) holds for the two functions above as well. Again, think of reflecting the entire image above over the line $y=x$ to see this.

$$\textstyle{\tan \theta = \cot (\frac{\pi}{2} - \theta) \quad \textrm{ and } \quad \cot \theta = \tan (\frac{\pi}{2} - \theta)}$$

Domain, Image/Range, Graphs, and "Inverses" for the Six Trigonometric Functions

Of course, with new functions introduced, we will want to know as much as we can about them. We need to ask all the standard questions:

  1. "What is the domain?" (assuming this is some subset of the reals, $\mathbb{R}$)
  2. "What is the related image/range?"
  3. "What does its graph look like?"
  4. "Does it have an inverse? ..and if not -- does it have any 'pieces' that are invertible?"

We aim to address all of these in this section. Let us address these in their natural pairs:

Some Useful Identities (Some Resulting From Complex Numbers!)

An identity is an equation whose left and right sides are equal for all values of the variables in their respective implicit domains. Some trigonometric identities (i.e., identities involving trigonometric functions) that prove useful in a great many contexts are given below, with a discussion of why each must hold. While all of these identities can be proven without appealing to complex numbers, some are proven far more easily with them!

Proving Other Trigonometric Identities

To prove that a trigonometric equation is an identity, one typically starts by trying to show that either one side of the proposed equality can be transformed into the other, or that both sides can be transformed into the same expression.

In other words, suppose $A$ and $B$ are some trigonometric expressions and we are trying to determine if $A=B$.

We hope that both expressions will simplify to some common form $C$, as if we can show the following:

$$\begin{array}{rclcrcl} A &=& A_1 \quad && \quad B &=& B_1\\ &=& A_2 && &=& B_2\\ &=& \cdots & \textrm{and} & &=& B_3\\ &=& A_n && &=& B_4\\ &=& C && &=& \cdots\\ & & && &=& B_m\\ & & && &=& C \end{array}$$

then we will know

$$A = A_1 = A_2 = \cdots = A_n = C = B_m = \cdots = B_2 = B_1 = B$$

and thus,

$$A = B$$

That is our general "plan of attack" -- although, we might get lucky and the sequence of $A_1, A_2, \cdots$ will terminate in $B$, or the sequence $B_1, B_2, \cdots$ will terminate in $A$, which then shortens our argument a bit.

There are some basic strategies to help us get to that common form $C$ as efficiently as possible:

Showing an Equation is Not an Identity

It may be the case that in the course of trying to prove a given equation is an identity, one begins to suspect that it is not.

In such situations, one should test whether the equation's left and right sides are actually equal by plugging in some values for the variables it contains. Remember, one only needs a single counter-example to prove an equation is not an identity.

If however, one tests a particular value (or set of values) and the left and right sides of the given equation agree in value, that particular test is inconclusive -- and a decision must be made whether to continue the search for a counter-example and test additional values, or to return to trying to prove the given equation is an identity.

To have the best chance of selecting values that will show a given equation is not an identity, one should keep the following in mind: