IntMath Newsletter – Trigonometry tips and a puzzle
1 Dec 2008
In this Newsletter:
1. Earth Puzzle
2. Math tip – Trigonometry
3. From the math blog
4. Solution
5. Final thought – 3 kinds of people
1. Earth Puzzle
I walk 20 paces North then turn left and walk another 20 paces West. I turn left again and walk a further 20 paces South. I am amazed to find that I’m back in the same place that I started.
Where am I?
[Hint: This puzzle is related to Earth Geometry that we talked about in the last edition. Solution at the end of this Newsletter.]
2. Math tip – Trigonometry
Benny is a subscriber to the IntMath Newsletter and he recently wrote:
I am going to a community college and will be taking trig next semester. So I would like to get a heads up on what I am getting into.
Well, Benny, you have taken a good first step by investigating what you are going to learn before the semester starts. Many students don’t start thinking about what they are learning until the first assignment is due — and then they have to scramble around and play catch up for the rest of the semester.
The word trigonometry is from Greek and it means “triangle measure”. So you’ll draw and study many triangles during your study of trigonometry, especially right-angled triangles.
Uses of Trigonometry
Let’s consider some of the uses of trigonometry in our everyday lives.
You will probably listen to some music today. The song you listen to has been recorded digitally (a process that requires Fast Fourier Transforms, which use trigonometry) and it has probably been compressed into MP3 format using lossy data compression (which uses an understanding of the human ear’s ability to distinguish between sounds — also requiring trigonometry).
Image Source
You will probably drive over a bridge today. That bridge was built using an understanding of forces acting at different angles. You will notice that bridges involve many triangles — trigonometry was used when designing the lengths and strengths of those triangles.
Image Source
Your car (or phone) may have an inbuilt GPS (Global Positioning System), that uses trigonometry to tell you exactly where you are on the Earth’s surface. It uses the data from several satellites and earth geometry like we learned about in the last IntMath Newsletter, then uses trigonometry to determine your latitude and longitude.
Image Source
On your way to school, you will pass a modern building. Before they built that structure, they needed to survey the area (using a leveling instrument) and then design the building (using 3-D modeling software), and determine the angle of the sun and winds (for best energy efficiency and placement of solar panels). All of these processes require an understanding of trigonometry.
Leveling instrument. Source
If you live near the sea, the tides affect what you can do at different times of the day. The tide charts that they publish for fishermen are predictions about tides years in advance. These predictions are made using trigonometry. Tides are an example of a periodic occurrence (they occur in repeating patterns. It’s not exactly periodic, but close.)
Image Source
In fact, trigonometry is important in almost all fields of science and engineering.
(See all the Uses of Trigonometry that are mentioned in Interactive Mathematics.)
What do you Learn in Trigonometry?
You usually start the study of trigonometry by looking at how right triangles are used to measure things that are otherwise quite difficult to measure. For example, heights of mountains and trees can be determined by the use of similar triangles. I can easily measure lengths AB and AC in triangle ABC (written ΔABC) and use that to find height DE. I could do a similar process to find the height of the mountain.
Image Source
What if the angles are different? Trigonometry allows us to use ratios that are associated with any angle ABC, so we can calculate a broad range of heights without having to measure them.
You will learn about three important ratios for any angle: sine (shortened to sin), cosine (cos) and tangent (tan). I strongly suggest that you learn these 3 ratios very well, since much of later trigonometry depends on them. (See Sine, Cosine, Tangent.)
Usually we measure angles using degrees (°) but these are not so useful for science and engineering. You will also learn about radians, which is an alternative — and more useful — unit for measuring angles. (See Radians.)
After you have mastered the basics, you will go on to learn about Graphs of Trigonometric Functions (think of the squiggles you see on an earthquake graph or a heart monitor) and then Analytic Trigonometry, which gives you a set of procedures that make it easier to solve more complex problems.
ECG of a 26 year-old patient. Source
Tips for Learning Trigonometry
a. Draw a lot: Drawing definitely helps with your understanding of trigonometry. When you need to solve problems later, it really is valuable if you can sketch the problem quickly and accurately. In particular:
- Draw the triangles that you are studying;
- Sketch the situation in the word problems; and
- Practice drawing the sine and cosine graphs until you can do it without having to join millions of dots on the page.
b. Learn the basics well: By “basics” I mean:
- The definitions of sin, cos and tan and how to use them in triangle problems;
- The signs of trig ratios of angles greater than 90° (i.e. know when they are positive or negative);
- The graphs of y = sin(x) and y = cos(x) (and the concept of periodic functions)
c. Take care using your calculator: The most common problems when using caculator in trigonometry include:
- Being in the wrong mode (e.g. being in degree mode when you should be in radian mode)
- Trusting the calculator more than your brain. The calculator will not always give you the correct sign (+ or −). Often you need to figure that out for yourself.
- Always estimate your answer first so you can check against what your calculator tells you.
- Make sure you know why your calculator should not use “sin-1” or “cos-1” on the buttons. This confuses many students and it is not necessary. We should use arcsin θ so it is not confused with 1/(sin θ)
So there you go Benny. I hope that gives you an idea of what trigonometry is used for, what it is about and what to watch out for. Sadly, trigonometry gets a bad press with many students. It doesn’t need to be so if you get on top of it early and follow the above tips.
3. From the math blog
1) Project Euler
Project Euler has some interesting math questions that require the use of computer algorithms to solve.
2) Unicode characters for Chinese and Japanese numbers
Unicode characters use hexadecimal numbers (base 16) to display characters from languages like Japanese, Chinese, and Greek.
3) Friday Math Movie – Math Rules!
This week’s math movie was a finalist in the X-Box Competition at the 2007 New York Television Festival.
4. Solution
The answer to the puzzle above is that I started at the South Pole. I walked 20 paces North, then 20 paces West, then 20 paces South again, arriving back at the South Pole, where I started.
In the picture, it looks like the second and third legs of the journey are not straight. This is one of the intriguing things about Earth Geometry — no lines are straight. (The first leg appears straight because North is straight up on the picture.)
4. Final thought – 3 Kinds of People
There are 3 kinds of people:
1. Not very clever people — the ones who never learn from their mistakes
2. Smart people who do learn from their mistakes
3. Successful people who learn from the mistakes of others
Which kind of person are you?
Until next time.
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- How to remember trigonometry ratios There's more than one way to remember the sine, cosine...
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1 Dec 2008 at 11:55 pm Link to this comment
Cientific American (SciAm.com) December 2008 Page 116
quote: ” Most colloquial descriptions of GPS say the technology uses triangulation to determine the position of a receiver on the earth. Mathematically speaking, the system uses trilateration. Triangulation determines position by measuring the angles of the triangles formed between an observer and three known points. Trilateration determines position by measuring the distance from an observer to three known points; the timing signals sent from GPS satelites (the known points) to a receiver determine those distances”.
Best wishes. Johan.
2 Dec 2008 at 1:34 am Link to this comment
Good letter.
2 Dec 2008 at 6:32 am Link to this comment
In reference to the recent newsletter on Trignometry-the kind of 3 persons. There are 3 persons in 1 sometimes, Mr. Murray. Some persons sometimes will learn, sometimes,they simply don’t see it good to work on the mistakes. However, people tend to dwell at a greater percentage of one of the 3 characteristics at times.
It was a challenge to me.
Thanks.
2 Dec 2008 at 12:10 pm Link to this comment
Johan: Thanks for the input.
Ed: Thanks!
Lani Joe: Yes, it is a challenge. It is sad that people seem to be programmed to repeat mistakes. We don’t learn as much as we should…
2 Dec 2008 at 5:23 pm Link to this comment
? wonder if you are going to prepare a page a bout derivative.?f you do so ? will be happy!
thanks for all.
3 Dec 2008 at 7:26 am Link to this comment
Very good, especially the last part…I generally find that its harder to get interested in math than actually do it.
Once you start actually enjoying it, you do so much better…
4 Dec 2008 at 1:36 am Link to this comment
Kinds of people: binary
There are 10 kinds of people in the world: those who can count in binary and those who cannot. And by the way, never trust a man who can count to 1024 on his fingers!
Kinds of people: ternary
There are only 10 types of people in the world —
those who understand ternary, those who don’t, and those who mistake it for binary.
Kinds of people: self-referential paradoxes
There are two kinds of people in the world: those who separate people into two kinds, and those who do not.
There are two kinds of people in this world: those who are good at math, those who are good at English, and those who ain’t good at neither.
There are two kinds of people, those who finish what they start and so on…
Dedication to Godel
There are two kinds of people in the world: those who get jokes, and those who don’t. Get it?
4 Dec 2008 at 7:54 am Link to this comment
r?fat: The derivatives topic is on my long list for future IntMath Newsletters. Watch this space!
Josh: Motivation is the key, for sure. As someone once said, “Every accomplishment starts with the decision to try.”
Maria: Thanks for your great list of dichotomies, trichotomies, quadrotomies — and so on…
4 Dec 2008 at 1:52 pm Link to this comment
you are doing good job keep it up
4 Dec 2008 at 3:25 pm Link to this comment
[...] Murray Bourne’s IntMath Newsletter this week includes a nice preamble to the study of Trigonometry. I’d like to be able to link to that item specifically when introducing the topic, so maybe I’ll ask him to isolate it if he has the time. [...]
4 Dec 2008 at 9:03 pm Link to this comment
Abdul: Thanks for the kind feedback.
Alan: I always have a dilemma – do I put an article like that in a separate post and then send all my subscribers there (which is one extra click of inconvenience and lots of readers will miss it) or do I include it in the body of the Newsletter (where it gets mixed up with the other stuff)?
I tell you what. This link will take your students directly to the beginning of the trigonometry part. Is that useful to you?
http://www.squarecirclez.com/blog/intmath-newsletter-trigonometry-tips-and-a-puzzle/1528#trig
5 Dec 2008 at 5:25 pm Link to this comment
I am very happy to see this.
11 Dec 2008 at 9:15 pm Link to this comment
HEY GUYS I REALLY LIKE THIS EDITION IT’S REALY INTERESTING PLEASE SEND ME MORE.THANK U
15 Jul 2009 at 4:08 pm Link to this comment
YOUR UNDERSTANDING AS WELL AS ABILITY TO EXPLAIN TRIGONOMETRY IS APPRECIABLE
19 Jul 2009 at 10:11 am Link to this comment
Thanks for your feedback, Deepti – glad you find it useful!
25 Oct 2009 at 12:57 am Link to this comment
i must thank the author of this site. this is undoubtedly a great site to learn mathematics. it makes me believe that math is not a fiction, but it takes us into pleasing and imaginary world of ficiton when you are able to unbutton its mystery.
Math has also its mystery, symphony and rythm, but unfortunately most of us fail to discover it. But i must say when you know the way of getting into it then it won’t give you anymore pinch. anyway thanks for the tips
28 Jul 2010 at 10:56 pm Link to this comment
Hi,
Love the post, but I disagree with your solution to your ‘Earth Puzzle’:
As the Earth can be approximated to being spherical, the poles are (all but) arbitrary points on a sphere (i.e. the poles, in terms of geometry, have no special significance). This means that your solution should work for /any/ point that I happen to choose on the surface of the Earth. I could, for example, go out into the street outside my house and give it a shot. It wouldn’t work.
For the experiment described to be effective, I’d have to walk further. A lot further. To get back to where I started after having made only 3 90 degree left turns (or 3 right ones), each side of the triangle would have to be equal in length to one quarter of the Earth’s circumference. This is because the effect of the Earth’s curvature becomes more pronounced on larger scales. Or, from the other direction, the smaller the scale you work on, the less pronounced the perceived curvature is (which is why the experiment won’t work with only 20 paces- unless you have one hell of an inside leg measurement- anywhere on the Earth).
Your experiment does work, however, if for the second side of the triangle you walk along a line of latitude. This line would be ‘straight’ in some sense, but not in the plane in which you were walking, so you would have to actually walk a curved path from your own point of view.
The rest of the post is excellent, though, and I’ll be bookmarking it for some ideas for the future!
31 Jul 2010 at 11:14 am Link to this comment
Hi TeaKayB. Thanks for your response, but indeed, my solution does involve the second side of the triangle being a parallel of latitude. This will happen if you start in 2 places – the North and South Poles (I chose the South Pole for my solution so I could “happen” to include my home country in my diagram
)
I can see why you want to only include the equator (the point reached when we travel 1/4 of the Earth’s circumference) as one of these latitude lines, but in fact, it works for any parallel of latitude.
A key point is what we mean by “turn 90° left”. In which plane is that 90 degrees measured? And what makes the Equator any more special for this measurement than the point 20 paces north of the South Pole?
I love spherical geometry!
31 Jul 2010 at 4:07 pm Link to this comment
What makes the equator special is that it is the only parallel of latitude on which the conductor of the experiment would be able to walk in, as far as they were concerned, a straight line, as they would be perpendicular to the plane of that parallel’s curvature.
Any other parallel would require the walker to consciously /not/ walk in a straight line in order to stay on the line of latitude. This is because anywhere other than the equator, they will not be standing/walking perpendicularly to the line’s curvature.
If we are restricting ourself to the 2-dimensional Earth’s surface when drawing our triangle, mine (that uses the equator and has sides of length 1/4 of the circumference) is as true a triangle you can get when including three right-angles – each of the three sides are “straight” as far as can be measured in that curved 2D space.
Your 20-pace triangle would not be a triangle on the 2D surface, as additional to being curved in the unseen/ignored 3rd dimension, one of the sides would also be demonstrably curved in the 2 dimensions that we’re working in.
Again, it is not the equator itself that is special; it is the size of the shape. The parallels of latitude are arbitrary (in terms of geometry) so if your assertion were true, it would work anywhere on the surface of the Earth. I could /only/ produce what appeared to me to be a 3-right-triangle if the sides were equal in length to 1/4 of the circumference of the sphere on which you’re drawing it.
Try it with a marker pen and a ball, or even a balloon- you /cannot/ draw a triangle with three right angles and sides longer or shorter than 1/4 circumference of the ball/balloon without introducing curvature other than the natural curvature of the body that you’re drawing it on.
31 Jul 2010 at 5:32 pm Link to this comment
OK, it’s confession time.
My puzzle question should have read “If you head North for 20 paces, then West for 20 paces, then South for 20 paces, you end up in the same place you started. Where are you?” The issue of 90° and curvature does not arise.
That’s what I intended in the first place and I realized my error after your first comment
, but couldn’t resist stirring the pot. Nobody else picked my error!
Thanks for alerting me. I have updated the post.
1 Aug 2010 at 4:46 am Link to this comment
Hahaha, yes, I’m a fellow pot-stirrer
That makes much more sense now!
Glad to be of service