A tale of two measures: degrees and radians

The transition from degrees to radians is often the most traumatic mathematical change that the student has to endure when moving from elementary to intermediate mathematics. The simplicity of 360° seems so much more welcoming than the equivalent of \(2\pi\) radians for a full circle. \(\pi\) is forbidding, because it is not the convenient fractional fiction \(\frac{22}{7}\), but rather a number which is both transcendental and irrational, and therefore somewhat “untidy”. Surely this tradeoff between simplicity and complexity must have been worth it, or it would not have been so ordained. Here we attempt to fathom the method in the madness.

What is an angle?

For most of us, the idea of an angle first arose when we studied geometry in elementary or primary school. We then encountered triangles, which are closed figures with three straight sides and three enclosed angles. An equilateral triangle is particularly symmetric, with three equal sides and three equal angles, as shown in Figure 1.

The point at which a line meets another line is called a vertex, which means a “turning point”. By convention, vertices (plural of vertex) are labelled with uppercase letters like \(A\), \(B\), and \(C\). The lengths of the sides opposite the vertices are assigned the lowercase labels \(a\), \(b\), and \(c\) respectively. The angles have been labelled with the Greek letters \(\alpha\), \(\beta\), and \(\gamma\). For all equilateral triangles, \(a = b = c\), by definition, and by symmetry, \(\alpha = \beta = \gamma\).

Degrees

On encountering geometry, we very likely proudly trotted out our set of mathematical instruments, which would include a pair of compasses, a protractor, one or two set squares, and a ruler or straight edge. Of these, the protractor—that plastic semi-circle marked out in degrees—was the proud badge that proclaimed that we had left behind arithmetic and progressed to geometry.

After we had learned to construct an equilateral triangle, using only compasses and a straight edge—without measurement by ruler—we would take out the protractor to verify that each angle of an equilateral triangle was indeed 60°. That small circle ° at the top—the superscript—was called the degree sign, and we could then jubilantly celebrate our first rite of passage into geometry and mathematical symbols.

Where did degrees come from?

Surely, degrees did not come from a protractor, although we use one to measure angles. How did degrees come about? With sixty degrees each in an equilateral triangle, ninety in a right angle , 180° in a straight line, and 360° in a full circle, how did degrees come to rule the roost of angular measure in elementary school?

Why not 100° in a full circle, or half circle, or even a quarter circle, also known as a right angle? Who imposed this measure upon us and what is its basis?

My favourite explanation for 360° degrees equalling a full circle is that the ancients estimated a solar year at around 360 days, and assigned one degree for each day of the year. Even if inexact, the number 360 had some sexagesimal¹ charm as it could be divided by the first three primes 2, 3, 5, and by their products. Indeed, 360 = 2³ × 3² × 5. Accordingly, 360 has a large family of factors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.

But beyond the approximation of a solar year, and the convenience of ready division by its factors, the use of degrees as a unit of angular measure is, to me, arbitrary. Who deemed the circle to be 360°, despite it being very factor-friendly? A less arbitrary measure of an angle was required for serious mathematics.

From triangles to circles

What is the root concept behind the idea of an angle? Harking back to the etymology of the word vertex—and applying it to the equilateral triangle—when one line changes direction by sixty degrees, we get the second line. These two lines form the angle. Therefore, a change of direction may also be called turning or rotation.

The quintessential two-dimensional geometric figure that is associated with rotation is of course the circle. It is the most simple and symmetrical two-dimensional figure we can construct. It is the path or locus traced out by a point that remains the same distance from a fixed point called the centre. When a protractor is centred on the centre of a circle, we can measure out degrees on the circumference of the circle. So far so good. But what about that magic number 360? Well, we are about to exorcise it now. 😉

Radians as an alternative to degrees

One traumatic transition for the student of elementary mathematics is when he or she is forced to abandon the warm comfort of degrees as angular measure, and compulsorily made to embrace the cold and cruel radian as the angular measure forever afterward. Why this unfair compulsion?

Using circles to measure angles

Because the idea of an angle is related to rotation, it seems natural that we should define angles using the circle as a basis, rather than the triangles that we encountered at first.

It is a fact that the length of a circle, or its perimeter, or its circumference, \(C\), is always related to its radius, \(r\), through the formula: \[ C = 2\pi r. \qquad{(1)}\] And \(\pi\) is not \(\frac{22}{7}\) as we were originally taught, but really a number whose precise expression cannot be predicted or exhausted. The digits simply keep rolling on, without pattern or end. But the beauty is that \(\pi\) is nevertheless a unique number, a universal mathematical constant. It seems that Nature has played a game on us by making the simple symmetrical circle have a circumference that can only be approximated but never entirely known to an unlimited precision.²

One radian

So, how does one define a radian? If, on the basis of its name, you guessed that it very likely involves the radius of a circle, your suspicion is well-founded. One radian is the angle subtended at the centre of a circle of radius one unit by an arc that is also one unit long. This is illustrated in Figure 2.

But what happens when our circle has a radius larger or smaller than one unit? We will take up this case, after a short detour.

Congruence and similarity

This is a mathematically non-rigorous digression on congruence and similarity, both of which are first encountered in school in the context of triangles, as shown in Figure 3.

Consider the triangle \(XYZ\), and ignore for a moment triangle \(LMN\). Suppose that \(XYZ\) is moved in the direction of the line \(PQ\) for a distance equal to the length of \(PQ\). We would then have the triangle \(LMN\).

Triangle \(LMN\), being a shifted version of triangle \(XYZ\), is identical with it, having identical respective angles and sides. Indeed, if triangle \(XYZ\) were laid on top of \(LNM\), we could not tell them apart. We say that triangle \(XYZ\) is congruent to triangle \(LMN\).

Any two-dimensional geometrical shape is congruent to another if the two shapes may be superimposed³ on each other to visually demonstrate that they are indistinguishable.

Similarity is less restrictive than congruence and applies to geometric objects that have the same shape but not necessarily the same size. In Figure 3, triangle \(ABC\) is similar to triangles \(XYZ\) and \(LMN\).

Intuitively, if two objects are similar, one may zoom in or zoom out on one object of the pair—without distortion—to obtain a version that may be superimposed on the other object to demonstrate that they are identical or congruent. In this case, we may enlarge triangle \(LMN\) until it attains the same size as triangle \(ABC\). It will then be congruent to \(ABC\).

The ratios of the respective lengths of corresponding sides of similar triangles are the same. In like fashion, the ratio of any arc length to the radius of a circle is the same for all arcs subtending the same angle at the centre. For example, the ratio of the circumference to the radius for two circles of radii \(r_1\) and \(r_2\) will be \(\frac{2\pi r_1}{r_1} =\frac{2\pi r_2}{r_2} = 2\pi\), which is a constant.⁴ This is a consequence of the fact that all circles are similar to each other.

What other classes of geometrical objects can you think of that are similar to each other within their class?⁵

Radians as angular measure

Consider Figure 4 in which two circles having different radii are shown.

We are now ready to define the value of any angle in radians. Consider a circle of radius \(r\). Let an arc of arbitrary length \(s\) subtend an angle of \(\theta\) at the centre. The angle \(\theta\) in radians is defined to be: \[ \theta \triangleq \frac{\text{arc length}}{\text{radius}} = \frac{s}{r}. \qquad{(2)}\]

By dividing the arc length by the radius, we have in effect normalized radian measure, and removed any trace of arbitrariness⁶ in its definition. And that is why we started out with Figure 2, which dealt with a circle of radius one unit. We know from Figure 4 that similarity guarantees that the value \(\frac{s}{r}\) for any given \(\theta\) is constant for all circles regardless of radius.

Note that the value of \(\theta\) is a ratio of two lengths and is therefore dimensionless in the sense of Physics. Although it may be considered a unitless pure number the SI units do define the radian as the SI unit of angular measure.

In summary, we have the following:

1. Radian angular measure is directly proportional to arc length on the circle.
2. This measure is independent of the radius of the circle.
3. The resulting “unit” is really a unitless ratio of two lengths.
4. Since 360° equals \(2\pi\) radians, 1° approximately equals 0.017453292 radians. Likewise, 1 radian equals approximately 57.29577951°.⁷

By defining radians as above, we remove the arbitrariness associated with 360° for a full circle. But the mathematical elegance and rigour conferred by radians comes at a cost. The angle of a full circle is \(2\pi\), which is a computationally inconvenient number to say the least.

If you think about it, the size of an angle in radians is expressed as a ratio of two lengths. But we have encountered angles being associated with ratios of lengths elsewhere in mathematics as well. Such ratios are familiar to us from trigonometry where the \(\sin\), \(\cos\), and \(\tan\) functions are expressed as the ratios of lengths in a right-angled triangle. We review this relationship next.

Trigonometric ratios

Trigonometric ratios are one of the workhorses of applied mathematics. They arose from the study of right-angled triangles. The three standard trigonometric ratios are called sine, cosine, and tangent. They are represented by the abbreviated functional names \(\sin\), \(\cos\), and \(\tan\) when used in mathematics. Figure 5 shows the pictorial definitions of these three trigonometric ratios. Notice particularly how these expressions are the unitless ratios of two lengths, just as with radians.

The Circular Functions

The trigonometric ratios in the context of triangles become the trigonometric functions in the context of the circle. They are then are also called the circular trigonometric functions, uniting the circle and the triangle as their progenitors. We will briefly review that relationship here, to better understand not only the terminology but also the hidden relationships between the triangle and the circle.⁸

I used to wonder why the word tangent was used for the name of a trigonometric function because a circle was not involved in its definition; a triangle was. But when the three standard trigonometric functions are viewed vis-a-vis a unit circle, the mystery behind the nomenclature is revealed.

The radian was introduced here using a unit circle. The same helpful unit circle will serve to relate the triangle and the circle to the trigonometric functions, as illustrated in Figure 6 below.

Figure 6 shows a unit circle drawn on the two-dimensional coordinate plane with \(x\) and \(y\) axes and grid markings. The centre of the circle is \(O\) and \(S\) is a variable point on the circumference of the circle, that makes a counter-clockwise angle \(\theta\) with the positive \(x\)-axis. As \(\theta\) varies, so does the position of \(S\) on the circle.

The line \(OS\) is a radius and therefore one unit in length. The perpendicular from \(S\) to the \(x\)-axis meets it at \(C\). Referring to Figure 5, we may say \(\cos\theta = \frac{OC}{OS} = OC\) since \(OS = 1\). Accordingly, the \(x\)-coordinate of \(S\) is \(\cos\theta\). Likewise, \(\sin\theta = \frac{SC}{OS} = SC\). Thus, the \(y\)-coordinate of \(S\) is \(\sin\theta\).

\(S\) is therefore the point with coordinates \((\cos\theta, \sin\theta)\).

Note that since the coordinates of \(S\) are confined to the unit circle, the values of \(\sin\theta\) and \(\cos\theta\) are confined to the closed interval \([-1, 1]\), i.e. they perforce have values lying between \(-1\) and \(1\), both inclusive. From Figure 6, we see that \((\cos\theta, \sin\theta)\)—which represent the coordinates of \(S\)—take on signed values in accordance with the signs of \(x\) and \(y\) in the respective quadrants. One could also view the associated lengths as signed values.

The tangent

The really insightful revelation from Figure 6 comes from looking at \(\tan\theta\). Have you ever wondered why the ratio \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) is called the tangent?

Recall from geometry that a straight line and a circle might lie relative to each other in three different ways as shown in Figure 7.

The line \(AB\) does not cut or intersect the circle at all. The line \(EF\) cuts the circle at two points, \(E\) and \(F\), and is called a chord. When the line \(EF\) moves parallel to itself toward the circumference of the circle, we get the chord \(E'F'\). Eventually the points \(E'\) and \(F'\) will coincide and the line will cut the circle at one and only one point.

This case is illustrated by the line \(CTD\) which cuts the circle at only one point, \(T\). \(CTD\) is called a tangent (line) to a circle and \(T\) is the point of tangency.⁹ Note that the radius \(OT\) is perpendicular to the tangent \(CTD\).

With that out of the way, we know from Figure 5, that the tangent is the ratio of the lengths of the opposite side to the adjacent side, which in Figure 6 translates to \(\tan\theta = \frac{SC}{OC}\). But the denominator in this case, \(OC\), is not \(1\) like it was for the other two trigonometric ratios. To work around this, with reference to Figure 6, we construct the triangle \(ODT\) thus:

1. Extend¹⁰ the line \(OC\) to intersect the circle at the point \(D\) which is \((1, 0)\). \(OD\), being a radius, has unit length.
2. Draw a tangent \(T'DT\) to the circle at \(D\).
3. Extend the radius \(OS\) to intersect the tangent at the point \(T\).

Because the triangle \(ODT\) is similar to triangle \(OCS\), we can assert that the ratios of corresponding sides are equal. Thus, \[ \tan\theta = \frac{CS}{OC} = \frac{DT}{OD} = DT \qquad{(3)}\] bearing in mind that, like \(OS\), \(OD\) is also a radius of unit length. We resorted to this construction for the following reasons:

1. Because \(T'DT\) is tangent to the circle, the angle \(ODT\) is a right angle.
2. The triangles \(OCS\) and \(ODT\) are therefore similar.
3. The length of \(OC\) is not one unit, but that of \(OD\) is one unit.

From Figure 6 the length of the tangent line \(DT\) is equal to \(\tan\theta\), explaining the nomenclature.

Therefore, the value of the tangent function for an angle \(\theta\) may be determined geometrically by extrapolating the radius \(OS\) until it intersects the tangent to the circle at \(D\) at a point called \(T\).¹¹ The length \(DT\) is the value of \(\tan\theta\).

Note, though, that \(\tan\theta\) is a length outside the unit circle and is therefore not constrained to take on values between \(-1\) and \(1\). Indeed, as \(\theta\) starts increasing in the first quadrant of the unit circle, you will notice that as \(OS\) approaches the \(y\)-axis and as \(\theta\) approaches \(\frac{\pi}{2}\) (or 90°, if you are still attached to degrees), the line \(OS\) is increasingly aligned with \(DT\). At \(\theta=\frac{\pi}{2}\), \(OS\) is parallel to \(DT\) and “never the twain shall meet”. Loosely speaking, parallel lines are only supposed to “meet at infinity” and that is why \(\tan\frac{\pi}{2}\) is said to be “infinite” at that point. I find this geometric explanation—of why \(\tan\theta\) does not assume a finite value at \(\theta=\frac{\pi}{2}\)—most fulfilling.

When \(\theta\) is in the third quadrant, for instance, \(OS\) extrapolated in the negative \(y\) direction will not intersect the tangent \(DT'\) in the negative \(y\) direction as they diverge. So, the line \(SO\) must be produced in the positive \(y\) direction to once more intersect the tangent \(T'DT\) at \(T\). That explains why tangents of angles in the third quadrant are positive.

The trigonometric functions

By moving from triangles to the unit circle on coordinate axes, we have enabled \(\theta\) to take on any value between 0 and \(2\pi\) radians. The trigonometric ratios have been unshackled from the triangle to become the trigonometric functions which can take on any real number as arguments. The graphs of the three standard trigonometric functions are shown in Figure 8.

Note that when \(\theta = 2\pi\) radians, we cannot really distinguish it from \(\theta=0\) radians. The trigonometric functions therefore repeat themselves every time the point \(S\) in Figure 6 completes a full circle: they are periodic with a period of \(2\pi\). So, one angle may masquerade as another unless we have accounting devices to optionally add \(2n\pi\) to it with the proviso that \(n\) is an integer.

When trigonometric functions like \(\sin x\) are differentiated or integrated, the variable \(x\), representing the angle, must be measured in radians. Moreover, when oscillatory phenomena, whether of water, or electricity, or light, are studied in the physical sciences and engineering, trigonometric functions usually form part of the solution. The importance of \(\pi\) is therefore entrenched in the importance of trigonometric functions and radian measure, in solving a variety of physical problems. And this concept is a segue to power series expansions of trigonometric functions, their use in calculus, and later on, in Fourier series.

Power series for trigonometric functions

This is where the plot really thickens.

Both the statements \(\sin(30°)=0.5\) and \(\sin(\frac{\pi}{6})=0.5\) are factually correct and perfectly acceptable. We will not be committing any mathematical heresies through either statement.¹²

But it is possible to express trigonometric functions in terms of power series in which an argument in degrees would be inadmissible. It is only after we cross this threshold in mathematics that radians truly come into their own, after which there is “no going back to the old ways”.

I will now do a bit of hand-waving and say that it has been proved¹³ that:

\[ \sin\theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots \qquad{(4)}\]

where the dots at the end of Equation 4 tell us to imagine that this series never ends but goes on forever following the pattern shown. Equation 4 demonstrates a paradox: a trigonometric function is not a polynomial; yet a trigonometric function may be expressed as an infinite polynomial. Infinity has this beguiling attribute of “enabling the impossible.”

Recall that if a number is less than one, raising it to a power greater than one makes it smaller than it originally was. So, when \(\theta\) is very close to zero, the higher powers on the right hand side (RHS) of Equation 4 become smaller and smaller, and may be ignored without much loss in accuracy. In this case, we may assert that:

\[ \sin\theta\approx\theta \:\text{for}\: \lvert\theta\rvert \to 0. \qquad{(5)}\]

In English this expression means that for vanishingly small values of \(\theta\)—whether positive or negative—\(\sin\theta\) is approximately equal to \(\theta\).

From Figure 5 we know that the number on the left hand side (LHS) of Equation 5 is a unitless ratio of two lengths and thus a “pure” number. This requires the right hand side to be also expressed in a similar unitless measure, and the radian fits the bill.

The validity of Equation 5 may also be seen from Figure 9 where the closeness of the curve \(f(\theta) = \theta\) and \(f(\theta) = \sin\theta\) near the origin is evident. Indeed, right up to a value of \(\lvert\theta\rvert\approx 0.3\), the two curves track each other closely.

Let us now illustrate the reasonableness of Equation 5 by evaluation. Set \(\theta\) to \(0.005\) radians, which is a small value close to zero. Then \(\sin\theta = \sin(0.005) = 0.004999979167\), which demonstrates the validity of Equation 5. However, if one were to interpret the number 0.005 as degrees rather than radians, we then have \(\sin(0.005°) = 0.00008726646249\) which is almost \(57\) times smaller than the number 0.005.

The moral of this example is that when we evaluate trigonometric functions in degrees in the context of their power series, we must apply a correction factor of \(\frac{\pi}{180}\) to implicitly convert the function argument on the LHS from degrees to radians. Otherwise, keeping the degree argument, we have to apply a factor of \(\frac{180}{\pi}\) to each term on the RHS. This is a layer of bookkeeping we may easily avoid by using radians on both sides of the equation.

Fourier series

Periodic waveforms repeat themselves indefinitely. The sinusoids—which are trigonometric functions like \(\sin\) and \(\cos\)—are one such example. Such waveforms arise frequently in signal analysis and synthesis in electrical engineering. In that context, the independent variable is time.

A periodic function with certain properties may be represented by an infinite sum of sinusoids. This was the great insight of Jean-Baptiste Joseph Fourier from whom Fourier series derive their name.

For example, let the original signal be a square waveform denoted by the function \(s(t)\) in Figure 10.¹⁴ Imagine that the single cycle of the square wave—shown below— is repeated periodically forever.

The Fourier series for such a square waveform is an infinite sum of sinusoids that collectively represent the waveform. This might seem like a tall order but it is nevertheless true. The Fourier series representation of the square wave \(s(t)\) is given by: \[ s(t) = \frac{2}{\pi}\left[\sin(t) + \frac{\sin(3t)}{3} + \frac{\sin(5t)}{5} + \frac{\sin(7t)} {7} + \cdots\right] \qquad{(6)}\] where again, the dots mean that the pattern repeats forever. Note that this is no approximation, but an equality.

The successive partial sums of the RHS of Equation 6 are termed \(s_1(t)\), \(s_2(t)\), etc., with \(s_1(t)\) denoting the first term, \(s_2(t)\), the sum of the first and second term etc. It is evident from Figure 10 that the more terms we add, the better the match between the original signal and its approximation, denoted generically by \(s_*(t)\) in the graph.¹⁵

On the surface, Equation 6 seems a remarkable claim. How could a square wave with right angle corners be the result of sums of sine waves which have no corners? The answer lies in the fact that the series goes on forever and “infinity confers the equality”.

Where do radians feature here? As with the power series for the sine function, it is on the RHS where the variable \(t\) should be expressed in radians. Note that while \(t\) may have units of time, the fact that radians are unitless does not intrude into the equation as an extraneous factor.

We will close with one more example where radians make mathematical life much easier.

Euler’s formula and identity

The prodigiously productive Swiss mathematician, Leonhard Euler, gave us many equations, one of which is known as Euler’s formula, shown below: \[ e^{i\theta} = \cos\theta + i\sin\theta \qquad{(7)}\] The letter \(i\) is called the imaginary unit and its definition as \(i^{2} = -1\), takes us into the field of complex numbers¹⁶.

When we substitute \(\theta=\pi\) into Equation 7, and transpose terms, we get what is called Euler’s identity: \[ \begin{aligned} e^{i\pi} &= \cos\pi + i\sin\pi\\ e^{i\pi} + 1 &= 0\\ \end{aligned} \qquad{(8)}\]

Equation 8 has been described as the most poetic mathematical equation because it unites in one equation the five most fundamental quantities in all of mathematics: \(e, i, \pi, 1\) and \(0\). And it could not have come about without radians. With that epiphany on beauty, we shall conclude our tale of two measures.

Acknowledgements

All illustrations in this blog were generated using the TikZ-PGF and PGFPlots packages with LaTeX and Pandoc. To the authors of these packages, and to others who posted numerous examples of their usage on the Web, my humble gratitude. For Figure 10, I have drawn upon and modified the example of caverac at the TeX StackExchange forum.

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