Trig without Tears Part 1:
Copyright © 1997–2023 by Stan Brown, BrownMath.com
Trig without Tears Part 1:
Copyright © 1997–2023 by Stan Brown, BrownMath.com
Trigonometry is fascinating! It started as the measurement (Greek metron) of triangles (Greek trigonon), but now it has been formalized under the influence of algebra and analytic geometry and we talk of trigonometric functions. not just sides and angles of triangles.
Trig is almost the ideal math subject. Big and complex enough to have all sorts of interesting odd corners, it is still small and regular enough to be taught thoroughly in a semester. (You can easily master the essential points in a week or so.) It has lots of obvious practical uses, some of which are actually taught in the usual trig course. And trig extends plenty of tentacles into other fields like complex numbers, logarithms, and calculus.
If you’d like to learn some of the history of trigonometry and peer into its dark corners, I recommend Trigonometric Delights by Eli Maor (Princeton University Press, 1998).
The computations in trigonometry used to be a big obstacle. But now that we have calculators, that’s no longer an issue.
Would you believe that when I studied trig, back when dinosaurs ruled the earth (actually, in the 1960s), to solve any problem we had to look up function values in long tables in the back of the book, and then multiply or divide those five-place decimals by hand? The “better” books even included tables of logs of the trig functions, so that we could save work by adding and subtracting five-place decimals instead of multiplying and dividing them. My College Outline Series trig book covered all of plane and spherical trigonometry in 188 pages—but then needed an additional 138 pages for the necessary tables!
Though calculators have freed us from tedious computation, there’s still one big stumbling block in the way many trig courses are taught: all those identities. They’re just too much to memorize. (Many students despair of understanding what’s going on, so they just try to memorize everything and hope for the best at exam time.) Is it tan² A + sec² A = 1 or tan² A = sec² A + 1? (Actually, it’s neither—see equation 39!)
Fortunately, you don’t need to memorize them. This paper shows you the few that you do need to memorize, and how you can produce the others as needed. I’ll present some ideas of my own, and a wonderful insight by W.W. Sawyer.
I wrote Trig without Tears to show that you need to memorize very little. Instead, you learn how all the pieces of trigonometry hang together, and you get used to combining identities in different ways so that you can derive most results on the fly in just a couple of steps.
You might like to read some ideas of mine on the pros and cons of memorizing.
To help you find things, I’ll number the most important equations and other facts. (Don’t worry about the gaps in the numbering. I’ve left those to make it easier to add information to these pages.)
A very few of those, which you need to memorize, will be marked “memorize”. Please don’t memorize the others. The whole point of Trig without Tears is to teach you how to derive them as needed without memorizing them. If you can’t think how to derive one, the boxes should make it easy to find it. But then please work through the explanation. I truly believe that if you once thoroughly understand how all these identities hang together, you’ll never have to memorize them again. (It’s worked for me since I first studied trig in 1965.)
This is rather a long document for reading on screen. If you prefer, you might want to print the printer-friendly version.
Trig is really a mix of practical and theoretical. On one hand, you find distances or angles in triangles; that’s pretty concrete. Computers do it now, but when artillery was guided by humans they used trig to figure how to aim the cannon. Land surveying still needs trig, and so do plenty of other fields. If you take a high-school physics course, you’ll use plenty of trig in it. Parts 2 through 4 in this book have the practical stuff.
But as you go further in math, though there are usually some real-world problems there somewhere, you spend a greater and greater proportion of your time manipulating symbols. That makes sense: tougher problems take more thought and more calculation to solve.
Traditionally, trig courses are a mix of the practical stuff with the stuff that has no direct practical application but that you’ll need to do work in calculus or physics or other technical fields later. Nearly all the trig identities fall into that category. Part 5 and afterward are the groundwork for those future courses.
I don’t say you should stop at the end of Part 4, even if you’re never planning to take another course. “Is it useful to me?” isn’t always the right question to ask. (If people decided things that way, video-game manufacturers would go out of business tomorrow.) Better questions are, “Is this interesting? Would I like to pursue this just for the fun of stretching my mind?”
Yes, even “Is it beautiful?” Symmetry is a lot of how we judge beauty—if you doubt that, picture someone good-looking, and then mentally change one side of their face. There are some beautiful symmetries in the first half of the book, but the second half has a lot more, plus answers to questions bright students have always asked, like What’s the logarithm of a negative number?
By the way, I love explaining things but sometimes I go on a bit too long. So I’ve put some interesting but nonessential notes at the end of most chapters and inserted hyperlinks to them at appropriate points. If you follow them (and I hope you will!), use your browser’s “back” command to return to the main text.
Much as it pains me to say so, if you’re pressed for time you can still get all the essential points by ignoring those side notes. In token of this, they’re labeled BTW. But you’ll miss some of the fun.
Trig without Tears deals exclusively with plane trigonometry, which is what’s taught today in nearly every first course. Spherical trigonometry is not part of this book.
I’m also restricting myself to real arguments to the functions and real values of the functions. I have to draw the line somewhere! (I do use real-valued functions with the polar form of complex numbers in the Notes.) For complex trigonometric functions, see chapter 14 of Eli Maor’s Trigonometric delights (Princeton University Press, 1998).
In Trig without Tears we’ll work with identities and solve triangles. A separate (and much shorter) page of mine explains how to solve trigonometric equations.
Let’s not take anything for granted. You probably remember some basic facts from earlier school, but—
In an hour, the minute hand of the clock travels all the way around the clock, through four right angles, so a full circle is 360°.
The famous Pythagorean Theorem or Theorem of Pythagoras, a² + b² = c², says that if you add the squares of the two legs you get the square of the hypotenuse.
If a triangle contains an obtuse angle (greater than 90° but less than 180°), it is called, naturally enough, an obtuse triangle. In a right triangle or obtuse triangle, the other two triangles must be acute; otherwise the three angles would total more than 180°.
Fractions other than ½ are written using the slash, such as a/b for a over b.
In talking about the domains and ranges of functions, it is handy to use interval notation. Thus instead of saying that x is between 0 and π, we can use the open interval (0, π) if the endpoints are not included, or the closed interval [0, π] if the endpoints are included.
You can also have a half-open interval. For instance, the interval [0, 2π) is all numbers ≥ 0 and < 2π. You could also say it’s the interval from 0 (inclusive) to 2π (exclusive).
Starting with Part 5, I’ll be referring to the four quadrants of a circle, or of the xy plane. The division for a circle would be the two lines from 12 to 6 and from 3 to 9; on the plane, the division is the x and y axes.
The quadrants are usually given Roman numerals, starting at the upper right and going counterclockwise. So Quadrant I or Q I is the top right quarter, Q II is the top left, Q III is the bottom left, and Q IV is the bottom right.
These pages will show examples with both radians and degrees. The same theorems apply to either way of measuring an angle, and you need to practice with both.
A lot of students seem to find radians terrifying. But measuring angles in degrees and radians is no worse than measuring temperature in °F and °C. In fact, angle measure is easier because 0°F and 0°C are not the same temperature, but 0° and 0 radians are the same angle.
Just remember that a complete circle is 2π radians. Then, think of the twelve hours numbered around the circumference of a clock face. When the hour hand goes all the way around, it travels through 2π radians or 360°. Six hours is half of that, 2π/2 = π or 180°, one hour is π/6 or 30°, two hours is 2π/6 = π/3 or 60°, and so on. (Thanks are due to Jeffrey T. Birt for this suggestion.)
One technical note: Angles don’t actually have units — they’re dimensionless. If you say “π radians”, you could just as well say “π” and leave off the “radians”.
If you’re measuring in degrees, then you do need to use the degree mark. 180° = π, or π radians. In other words, that degree mark (°) just means “×π/180”. It’s the same sort of animal as the percent sign (%), which really just means “/100”.
So you can convert between degrees and radians exactly the same way you convert between inches and feet, or between centimeters and meters. (If conversions in general are a problem for you, you might like to consult my page on that topic.)
But even though you can convert between degrees and radians, it’s probably better to learn to think in both. Here’s an analogy.
When you learn a foreign language L, you go through a stage where you mentally translate what someone says in L into your own language, formulate your answer in your own language, mentally translate it into L, and then speak. Eventually you get past that stage, and you carry on a conversation in the other language without translating. Not only is it more fun, it’s a heck of a lot faster and easier.
You want to train yourself to work with radians for the same reason: it’s more efficient, and saving work is always good. Practice visualizing an angle of π/6 or 3π/4 or 5π/3 directly, without translating to degrees. You’ll be surprised how quickly it will become second nature!
To get the most benefit from these problems, work them without first looking at the solutions. Refer back to the chapter text if you need to refresh your memory.
Recommendation: Work them on paper — it’s harder to fool yourself about whether you really understand a problem completely.
You’ll find full solutions for all problems. Don’t just check your answers, but check your method too.
Where possible, give an exact answer rather than a decimal approximation.
Dad sighed. “Kip, do you think that table was brought down from on high by an archangel?”
Robert A. Heinlein, in Have Space Suit—Will Travel (1958)
It’s not just that there are so many trig identities; they seem so arbitrary. Of course they’re not really arbitrary, since all can be proved; but when you try to memorize all of them they seem like a jumble of symbols where the right ones aren’t more obviously right than the wrong ones. For example, is it sec² A = 1 + tan² A or tan² A = 1 + sec² A ? I doubt you know off hand which is right; I certainly don’t remember. Who can remember a dozen or more like that, and remember all of them accurately?
Too many teachers expect (or allow) students to memorize the trig identities and parrot them on demand, much like a series of Bible verses. In other words, even if they’re originally taught as a series of connected propositions, they’re remembered and used as a set of unrelated facts. And that, I think, is the problem. The trig identities were not brought down by an archangel; they were developed by mathematicians, and it’s well within your grasp to re-develop them when you need to. With effort, we can remember a few key facts about anything. But it’s much easier if we can fit them into a context, so that the identities work together as a whole.
Why bother? Well, of course it will make your life easier in trig class. But you’ll also need the trig identities in later math classes, especially calculus, and in physics and engineering classes. In all of those, you’ll find the going much easier if you’re thoroughly grounded in trigonometry as a unified field of knowledge instead of a collection of unrelated facts.
This is why it’s easier to remember almost any song than an equivalent length of prose: the song gives you additional cues in the rhythm, common patterns of emphasis, and usually rhymes at the ends of lines. With prose you have only the general thought to hold it together, so that you must memorize it essentially as a series of words. With the song there are internal structures that help you, even if you’re not aware of them.
If you’re memorizing Lincoln’s Gettysburg Address, you might have trouble remembering whether he said “recall” or “remember” at a certain point; in a song, there’s no possible doubt which of those words is right because the wrong one won’t fit in the rhythm.
I’m not against all memorization. Some things have to be memorized because they’re a matter of definition. Others you may choose to memorize because you use them very often, you’re confident you can memorize them correctly, and the derivation takes more time than you’re comfortable with. Still others you may not set out to memorize, but after using them many times you find you’ve memorized them without trying to—much like a telephone number that you dial often.
I’m not against all memorization; I’m against needless memorization used as a substitute for thought. If you decide in particular cases that memory works well for you, I won’t argue. But I do hope you see the need to be able to re-derive things on the spot, in case your memory fails. Have you ever dialed a friend’s telephone number and found you couldn’t quite remember whether it was 6821 or 8621? If you can’t remember a phone number, you have to look it up in the book. My goal is to free you from having to look up trig identities in the book.
Thanks to David Dixon for an illuminating exchange of notes on this topic. He made me realize that I was sounding more anti-memory than I meant to, and in consequence I’ve added this note. But he may not necessarily agree with what I say here.
I wrote these pages to show you how to make the trig identities “fit” as a coherent whole, so that you’ll have no more doubt about them than you do about the words of a song you know well. The difference is that you won’t need to do it from memory. And you’ll gain the sense of power that comes from mastering your subject instead of groping tentatively and hoping to strike the right answer by good luck.
next: 2/Six Functions