CHAPTER 7

From Triangles to Seesaws

The key source of the wormhole-like surprise of Euler’s formula is its imaginary-number exponent, i times π. Euler was the first to figure out how to interpret such strange exponents.

As we’ve seen, in the mid-1700s many mathematicians still regarded the imaginaries as iffy numbers—Euler himself said that they had an air of impossibility. Thus, imaginary exponents really pushed the envelope in math at the time. The very idea of raising a number to an imaginary power may well have seemed to most of the era’s mathematicians like asking the ghost of a late amphibian to jump up on a harpsichord and play a minuet.

But Euler loved pushing envelopes. As math pioneers often do, he would manipulate well-accepted concepts and symbols to derive novel equations, then use the novelties to derive further math- and mind-expanding results. Using this strategy he showed that imaginary exponents could be translated into unexpectedly familiar terms.

Though wonderfully ingenious, the way he did that (which led to e^iπ + 1 = 0) isn’t hard to follow for those who have taken high school calculus. I’ll give you a simplified version of it a bit further on, sans calculus.

But first a preview: Euler showed that e raised to an imaginary-number power can be turned into the sines and cosines of trigonometry.

Before you sue me for breach of contract (“It was fraud plain and simple, Your Honor—he promised easy math and the next thing I knew I was choking on trig.”), let me offer you a no-choke mini-primer on sines and cosines.

Sines and cosines are functions. As noted earlier, functions resemble computer programs that take in numbers, manipulate them in some defined way, and output the results. But the trig functions are more interesting than simple functions such as 2x + 8. They’re like automated phonebooks that use people’s names as input and output their phone numbers by looking them up in directories consisting of names paired with the phone numbers. (The conceptual roots of this analogy go back to, you guessed it, Leonhard Euler. He originated the fruitful idea of thinking about functions in general terms as coupled numbers. As he put it, functions are essentially quantities that “so depend on other quantities that if the latter are changed the former undergo change.”)

The trig functions’ input consists of the sizes of angles inside right triangles. Their output consists of the ratios of the lengths of the triangles’ sides. Thus, they act as if they contained phone-directory-like groups of paired entries, one of which is an angle, and the other is a ratio of triangle-side lengths associated with the angle. That makes them very useful for figuring out the dimensions of triangles based on limited information. The word trigonometry comes from the ancient Greek for “triangle measurement.”

To understand how the trig functions work, you need to know that a right triangle, as pictured in Figure 7.1, is one with a 90-degree angle and two smaller angles. The side opposite the right angle is called the hypotenuse. I’ve arbitrarily assigned the hypotenuse here a length of 1 unit. The units could be any measure of distance—millimeters, inches, miles, light-years, you name it.

FIGURE 7.1

L_o stands for the length of the side that’s opposite the angle marked θ, and L_a represents the length of the side adjacent to the angle θ. (The side that’s not the hypotenuse, that is.)

The sine function is usually abbreviated as sin θ, where the Greek symbol θ (theta) is a variable representing the size of an angle. Note that θ has multiple roles here: it stands for the name of an angle as well as its size. Because it’s a variable, you can plug in numbers for it in sin θ, and the function, in effect, will output other numbers in a precisely determined way. A typical input number would be the size, in degrees, of one of the two non-90-degree angles inside a right triangle. The sine function would then output the ratio between the length of the side opposite to that angle and the length of the hypotenuse.

For the triangle shown above, the sine of the angle θ, sin θ, equals the ratio of L_o to the length of the hypotenuse, which is 1. Expressed as a fraction, that’s L_o /1, and since any fraction with a denominator of 1 can be reduced to its numerator alone (examples: 4/1 = 4 and 200/1 = 200), we have sin θ = L_o /1 = L_o. If we turn these stuck-together equations around and drop out the L_o /1 part, we get L_o = sin θ. This last equation means that the sine of the angle θ in the above triangle will tell us the length of the side marked L_o.

Now let’s plug in a specific angle measurement for θ. The angle θ in the above triangle is about 38 degrees. My calculator says that the sine of that angle is approximately 0.616. (Calculators express fractions—which, recall, are conceptually equivalent to ratios—as the fractions’ decimal equivalents.)

Therefore, we have L_o = sin 38^o ≈ 0.616 units. (The squiggly equals sign in front of 0.616 means “approximately equal to.”) Note that the sine function (that is, the version of it that exists in my calculator) let me determine L_o without the use of a ruler. This is an example of how trig functions can help size up a right triangle’s dimensions based on limited information, which in this case was the size of its angle θ and the one-unit length of its hypotenuse.

The cosine function, written cos θ, is similar to the sine function, except that it outputs the ratio between the length of an angle’s adjacent side and the length of the hypotenuse. Thus, for the triangle shown above, cos θ = cos 38^o = L_a/1 = L_a. Calling on my calculator again, I found out that cos 38^o ≈ 0.788, and thereby determined that the length of the side adjacent to the angle θ is about 0.788 units.

Here’s a simple exercise to help you get these trig definitions down. Draw a right triangle with a 1-foot-long hypotenuse on a standard 8½-by-11 sheet of typing paper. (You’ll need to draw the hypotenuse near a diagonal between two of the sheet’s corners in order to make it fit.) Use a protractor to measure one of the triangle’s non-90-degree angles. (Don’t have a protractor? Here’s a work-around: if you draw the triangle’s hypotenuse exactly on top of the sheet’s diagonal, the angle between it and a triangle side drawn parallel to the sheet’s shorter side will be about 52.3 degrees.) Then, following the procedure sketched above, use the trig functions to predict the lengths of the triangle’s two sides (the ones other than the hypotenuse) in inches.^*

To complete the exercise, measure the two triangle sides with a ruler to check whether your trig-based predictions are about right.

SINCE I’VE SPECIFIED plugging right triangles’ angles into the sine and cosine functions, you might think that inputting angles greater than 90 degrees wouldn’t be allowed. Indeed, if the angle θ in the above triangle were 90 degrees or more, there wouldn’t even be a triangle to talk about.

But there’s a clever way to expand the definitions of the sine and cosine functions to permit input angles greater than 90 degrees. As I’ll explain, this change can be thought of as reprogramming the two functions so that their internal directories are based on angles swept out within a circle, rather than on angles within triangles.

To see how the expanded definitions work, you need to know a little about the xy plane, which sprang from the work of seventeenth century French philosopher and mathematician René Descartes. It’s usually pictured as a flat, two-dimensional surface featuring a horizontal number line called the x axis and a vertical number line called the y axis.

Points on the plane, as shown in Figure 7.2, are specified by pairs of numbers inside parentheses, called x and y coordinates. The coordinates are used to locate points in much the same way that street intersections can be pinpointed on a map with numbered east-west streets and north-south avenues. For the coordinate pair (2,3), 2 is the x coordinate. It specifies how far from the y axis the point represented by (2,3) is and thus stands for a horizontal distance marched off along the x axis. The coordinate pair’s second number, 3, is the y coordinate—it shows how far the point designated by (2,3) is from the x axis and so represents a distance marched off vertically along the y axis. The point where the axes meet is called the origin, and its coordinate pair is (0,0).

FIGURE 7.2

As you may remember from math classes, equations such as y = 2x or y = x² can be represented as lines or curves on the xy plane by plotting points whose x and y coordinates satisfy the equations. This enables picturing functions geometrically, which often reveals things about them that would otherwise be hard to perceive. But that’s analytic geometry, and we have trig fish to fry here, so let’s get back to angles. Instead of pondering ones within triangles, however, we’ll consider angles within a special circle at the center of the xy plane. It’s called the unit circle.

(Historical aside: Trig was largely pioneered by astronomers who wanted to use it to determine distances to far-off objects such as the moon, and ancient Greek astronomer Hipparchus of Rhodes is credited with originating several of its key concepts, including the idea that lengths of chords, lines drawn within a circle, are related to the lengths of arcs measured along its circumference—a forerunner to defining trig functions in terms of circles. But it was Euler who put the unit circle front and center in trig, and his doing so crystalized what mathematicians regard as the modern version of trigonometry. In the remainder of this chapter, I’ll give you a simplified description of this version.)

FIGURE 7.3

The center of the unit circle is the point (0,0), the origin, and its radius is one unit in length. As shown in Figure 7.3, the unit circle intersects the x and y axes at the points (1,0), (0,1), (−1,0), and (0, −1).

A one-unit-long line segment akin to a clock’s minute hand is usually drawn inside the circle to indicate the sweeping out of angles. Looking at the diagram, you should picture this segment initially positioned with its tip at the point (1,0)—that is, as if it’s pointing at three o’clock—and then rotating counterclockwise around the origin. This rotary motion sweeps out an angle θ, as shown.

After sweeping out θ, the end of the segment is positioned at a point lying on the unit circle whose coordinate pair in the diagram is (L_a, L_o)—meaning that to locate the point, you measure L_a units horizontally along the x axis, and L_o units vertically along the y axis. You’ve seen L_a and L_o before: they represent the same lengths shown in the triangle above. In fact, that same triangle is delineated within the circle, and its one-unit-long hypotenuse is represented by the one-unit-long angle-sweeping segment. The triangle also includes the angle θ that was pictured earlier.

I’ve drawn the previously featured triangle within the circle to help show you how the triangle-based definitions of the trig functions can be recast in terms of the unit circle. The details on how that’s done are explained in the next few paragraphs. Believe it or not, you’re already an old hand at such mapping of math ideas from one context to another. For instance, when learning about angles, you mapped the idea of numbers onto the concept of angular distances between two intersecting lines—you probably didn’t even notice yourself making this rather impressive conceptual leap.

The triangle’s side along the x axis, which is L_a units in length, is adjacent to the angle θ of the triangle, just as it was in the earlier triangle diagram. Thus, cos θ = L_a/1 = L_a, based on the triangle-based definition of the cosine function. (This double equation should look familiar—it appeared above with the triangle.)

But L_a now has a second meaning, one that’s related to the unit circle. That is, it designates the distance that must be marched off along the x axis to specify the x coordinate of the point on the circle at the tip of the angle-sweeping line segment. And because the two-part equation cos θ = L_a/1 = L_a tells us that L_a is equal to cos θ, we know that this point’s x coordinate can be written as cos θ instead of L_a.

Similar logic applies to the y coordinate of the point at the tip of the angle-sweeping segment. That is, sin θ = L_o/1 = L_o, which means that sin θ can be used as a proxy for L_o when writing the point’s coordinate pair.

Therefore, as indicated in the diagram, the point’s coordinate pair, (L_a, L_o), can also be expressed as (cos θ, sin θ).

Now let’s add some action to the scene. If the angle sweeper were initially positioned with its tip at the three o’clock position and then rotated counterclockwise, its tip could effectively single out any point on the unit circle between the three o’clock and twelve o’clock positions. Each such point would be associated with a certain angle, call it θ, between 0 and 90 degrees. And for each such point, you could draw a right triangle inside the circle containing an angle θ along with a one-unit-long hypotenuse identical to the angle sweeper. For instance, if θ were close to 90 degrees, the triangle you’d draw would be tall and skinny, with L_a approaching a length of 0 units, and L_o approaching one unit. And for that tall, skinny triangle with a hypotenuse that’s one unit long, the definitions of the cosine and sine functions would guarantee that cos θ = L_a and sin θ = L_o.

Conclusion: the coordinates of the point at the sweeper’s tip can always be expressed as (cos θ, sin θ) when θ is between 0 and 90 degrees.

Notice that we have just opened up a new possibility for evaluating the cosine and sine functions for angles between 0 and 90 degrees: instead of outputting triangles’ side-length ratios, the functions can output the x and y coordinates of points along the unit circle at the angle sweeper’s tip after the sweeper designates an angle of θ degrees. In fact, if we redefined the functions so that their internal directories, so to speak, consisted of angles paired with these x and y coordinates (for cosines and sines, respectively), we wouldn’t need to change the numbers in the functions’ hypothetical directories at all. As we’ve just seen, they would contain the same input-to-output pairings regardless of whether they were based on right triangles or coordinate pairs along the unit circle.

BUT WHAT WOULD THE TRIG functions output if the angle sweeper got carried away and swept out an angle greater than 90 degrees?

This question can’t be answered with reference to the kind of right triangle inside the unit circle that’s shown above. The right triangle we’d need would have one right angle and another angle, call it θ again, larger than 90 degrees. There’s no such triangle.

But fortunately our new circle-based alternative for specifying the functions’ output can deal with this situation. That is, we can redefine the trig functions so that their output is based on the x and y coordinates of the point at the angle sweeper’s tip after it sweeps out an angle greater than 90 degrees. (As well as less than 90 degrees.) And if we assume that the angle sweeper is moving in the negative direction when it rotates in a clockwise direction (like moving left of zero along the number line), the redefinition will even allow negative angles to be plugged into the sine and cosine functions. Who needs triangles?

Consider an example of the redefinition at work: Assume the angle sweeper has swept out a 180-degree angle, a half circle. Since by convention it always starts at the three o’clock position, it is now pointing in the nine o’clock direction—that is, at the point with coordinates (−1,0). (Take a look at the diagram above if you have trouble picturing this.) Therefore, under the new definitions of the trig functions, the cosine function’s output when that angle is plugged in must be −1, and the output of the sine function for the angle must be 0. If we write that with equations, we have cos 180^o = −1 and sin 180^o = 0. This example, as you’ll later see, is crucial to understanding Euler’s formula.

Here’s another mini-exercise: What are cos 90^o and sin 90^o equal to? How about cos 360^o and sin 360^o?^*

Recapitulating, we’ve entered a triangle-free trig zone in which the cosine and sine functions have been souped up to handle any angle as input. (This includes angles greater than 360 degrees, as I’ll explain momentarily.)

Figure 7.4 shows the souped-up definitions applied to an angle between 90 and 180 degrees. Note that in this case cos θ must be a negative number between 0 and −1. That’s because after the sweeper has swept out the angle, the x coordinate of the point at its tip is between 0 and −1, and, of course, cos θ is defined as the value of that coordinate. Similarly, sin θ for angles in that range is between 0 and 1, as I hope is apparent from the diagram. (If it’s not, consider the tip point’s y coordinate in light of where the point would be on the xy plane when the angle θ is in that range.)

FIGURE 7.4

Please try yet another exercise: determine the possible range of values for cos θ and sin θ when θ is greater than 180 degrees (a half circle) but less than 270 degrees (three-quarters of a circle), and when θ is greater than 540 degrees and less than 630 degrees.^*

Hint on the second exercise with big angles: Sweeping out 360 degrees requires the angle sweeper to rotate all the way around the unit circle one time. And if it swept out, for example, 450 degrees, it would travel all the way around, back to where it started, plus 90 degrees beyond that point, because 450 = 360 + 90. Thus, after sweeping out 450 degrees, the sweeper is positioned exactly where it would be if it had swept out only 90 degrees. It follows that the sine and cosine of 450 degrees are the same as the sine and cosine of 90 degrees. (That’s because the trig functions’ output for both angles is determined by the coordinates of the same point on the unit circle.)

 Similarly, the sine and cosine of any angle will be the same as the sine and cosine of an angle that’s 360 degrees less—or 2 × 360 degrees less, or 3 × 360 degrees less, or n × 360 degrees less for any integer value of n. This implies that the sine and cosine of any angle θ greater than 360 degrees is equal to the sine and cosine of a readily visualized angle between 0 and 360 degrees, and you can calculate the latter angle by repeatedly subtracting 360 degrees from θ until the result is between 0 and 360 degrees. The fact that trig functions cyclically output the same numbers as ever larger angles are plugged into them, by the way, explains why endlessly undulating “sinusoidal” curves are produced when the sine and cosine functions are graphed in the xy plane.

If you had lots of time and a couple of measuring instruments (a protractor and a ruler), you could compile a table of the sine and cosine functions’ outputs when various angles are plugged into them. (This table, by the way, could serve as a somewhat crude directory for looking up the outputs when angles are fed into the functions.) To put it together, you’d use the protractor to draw the angle sweeper inside a unit circle on the xy plane at precisely determined angles between 0 and 360 degrees (each of which would be swept out, as always, beginning at the three o’clock position). If you were very exacting, you might do this by one-degree increments around the entire circle. Then for each angle you’d determine the coordinates of the point at the segment’s tip by carefully measuring the point’s distances from the x and y axes. (You’d measure the distances along lines traced out between the points and the axes so that the lines intersected the axes at right angles.) Finally, you’d record the coordinates in your table as the values for the cosine and sine functions for that angle.

After you’d completed the table, you could use it to position the angle sweeper to indicate a particular angle—say 143 degrees—without using a protractor. To do that, you’d look up the values of cos 143^o and sin 143^o in your table in order to use them as the coordinates of the point that should be at the sweeper’s tip after it sweeps out 143 degrees. Since cos 143^o is equal to about −0.799, you’d measure 0.799 units in the negative direction (left from zero) on the x axis. And since sin 143^o equals about 0.602, you’d measure that distance from the origin along the y axis. Based on these measurements, you’d make a dot at the point whose coordinates are (cos 143^o, sin 143^o), or (−0.799, 0.602). (It should be on the unit circle.) Finally, you’d draw the angle-sweeping line segment extending from the origin to the dot. The segment, as drawn, would reveal just how big the angle is.

One way to think about all this is to picture the coordinate pair (cos θ, sin θ) as a kind of device to automate the sweeping out of angles inside the unit circle. When you plug in an angle for θ, (cos θ, sin θ) makes the sweeper wind up exactly where it should be after rotating through θ degrees, by specifying the position of its tip point after it sweeps out that angle.

The redefined trig functions can be used to model oscillation (see box on next page), such as the cyclic movement of swings and seesaws (small kids are really into oscillation). The alternating current emanating from every plug in your home is also oscillatory, and electrical engineers have used gobs of trig since the late 1800s when designing AC-based circuits. By rights, a lot of the electrical devices in our lives should have little signs on the back that read “Trig Inside.”

To put the finishing touch on this minimalist trig primer, I’ll briefly introduce an alternative to degrees for measuring angles: radians. We need radians because angles plugged into the redefined sine and cosine functions (based on the unit circle) are usually measured in radians rather than in degrees.

Why radians? The reason is that specifying angles in radians often makes calculations easier than when degrees are used. The ancient Babylonians originated the use of degrees to measure angles—they liked the number 60 and its multiples (such as 6 × 60, or 360) so much that they based all their math on it. (It seems they were inspired by the fact that there are about 360 days in a year.) Thus, measuring angles with degrees, while often handy, is actually a vestige of an ancient way of thinking, and it entails needless clutter and complexity when used in higher math (and even in some lower math contexts)—degrees are like Roman numerals in this regard.^* These hassles can be avoided by using radians.

Oscillation is broadly defined as back-and-forth motion. To see how oscillation is linked to trig functions, picture the angle sweeper inside the unit circle continually rotating like a clock’s minute hand. With this image held in your mind, note that the sweeper is going back and forth between the three o’clock and nine o’clock positions as time passes. (Or between any two diametrically opposed points along the circle, for that matter.) This going back and forth is a form of oscillation, which means that oscillation can be represented in terms of rotary motion. Now imagine that the rotation is driven by plugging continually larger angles into θ in the coordinate pair (cos θ, sin θ)—the driver, so to speak, of the angle sweeper within the unit circle. If θ were increased by, say, 360 degrees per second, the point at the sweeper’s tip would travel around the circle once a second. You could think of this trig-function-driven motion as oscillation at one cycle per second.

Historians credit Galileo with conceptually linking rotary motion to oscillation. But even before the great Italian scientist saw how the two are conceptually linked, anonymous sixteenth-century German inventors put the linkage to use in the first treadle-driven spinning wheels, which translate oscillatory up-and-down foot motion into rotary motion.

Radians fit hand-in-glove with the sweeping out of angles within the unit circle. When an angle sweeper moves along the circle to sweep out an angle, the point at its tip moves a certain distance along the circle. Radians are based on this arc length. Specifically, a radian is defined as the angle swept out by an arc along the circle that’s equal in length to its radius. In the special case of the unit circle, the length of the radius is 1, and so the arc length associated with an angle of 1 radian is 1 unit.

How many radians would be swept out if the angle sweeper traveled all the way around the unit circle one time (that is, 360 degrees)? To get the answer, recall that the circumference of any circle is equal to its diameter times π. Since a circle’s diameter is twice as long as its radius, the circumference equals 2 × r × π units, where r stands for the length of its radius, which is simply 1 for the unit circle. Therefore, the angle in terms of radians in this case would be 2 × 1 × π, or, more compactly, 2π radians. Knowing that π is about 3.14, we could take this calculation one step further to conclude that the angle would be about 2 × 3.14, or 6.28, radians. But this step is very rarely taken in math—when radians are used to measure angles, they are almost always specified in terms of numbers multiplied times π.

Now that we’ve established that 360 degrees equals 2π radians, we know that half of 2π radians, or π radians, must be equal to half of 360 degrees, or 180 degrees. Similarly, half of π radians (180 degrees), or π/2 radians, must be equal to half of 180 degrees, or 90 degrees.

In sum, we have 2π radians = 360^o, π radians = 180^o, and π/2 radians = 90^o. This is all you need to know about radians. (If you have trouble remembering these radian/angle equivalences, just glance at the unit-circle illustration on the book’s cover.)

As we saw earlier, cos 180^o = −1 and sin 180^o = 0. Since π radians = 180^o, this implies that cos π = −1 and sin π = 0. (The word radians is typically omitted when writing angle sizes in trig functions.) These two trig facts will come into play when I show you how to bring forth Euler’s formula.

Four other trig facts will also come up later. By examining the coordinates of the point at the sweeper’s tip after it sweeps out π/2 radians (90°), you can see that cos π/2 = 0 and sin π/2 = 1. Lastly, cos 0 = 1 and sin 0 = 0, which follow from the fact if the sweeper moves zero radians, its tip remains at the point whose coordinates are (1,0).

IF YOU’VE BEEN ABLE to handle this chapter, you’re ready to take possession of the once-hidden connection between trig functions and imaginary-number exponents that Euler uncovered. This connection quickly leads to Euler’s formula and its fascinating implications. But before going into its details, in the next chapter I’ll show you how trig functions, imaginary numbers, and infinity can be combined in a readily understood way to arrive at a pleasing result. It will introduce you to the kind of conceptual moves that Euler made as he explored the hidden trails leading to e^iπ + 1 = 0.

* Some helpful hints: If you don’t have a calculator to look up the sine and cosine of an angle, you can use Google instead. Simply enter something like “sin 72 degrees =” (without quote marks) as the search term and the search engine will act like a calculator. If your triangle’s hypotenuse is 1 foot long, the side lengths you calculate with the trig functions will be expressed in the same units, namely feet. To convert these lengths to inches, multiply them times 12. Then, when you measure the triangle sides with a ruler, convert fractions of an inch to their decimal equivalents so that both the trig-predicted lengths and the ones you measure with a ruler are expressed in decimals. To do the conversion you simply divide the denominator of each fraction into its numerator. (A fraction can be regarded as a little division problem in which the denominator is divided into the numerator to calculate the fraction’s decimal equivalent.)

* Answers: cos 90° = 0, sin 90° = 1, cos 360° = 1, sin 360° = 0.

* Answer: In both cases, cos θ and sin θ will be between 0 and −1.

* The militaristic, bullying Romans were so hopelessly inert at math that if the ancient world had had a no-civilization-left-behind program in mathematics they would have been stuck in the remedial category for well over a thousand years. As math historian Morris Kline put it, their “entire role in the history of mathematics was that of an agent of destruction.” Most notably, they murdered Archimedes, the incandescent, irrepressible, eureka-shouting Feynman of the ancient world. At least there weren’t nuke codes in those days for their Neros and Caligulas to get their hands on.