## What is trigonometry all about?

[13 Jun 2011]

Benny is a subscriber to the IntMath Newsletter. 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 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.

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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 this IntMath Newsletter, then uses trigonometry to determine your latitude and longitude.

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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) which also requires trigonometry.

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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.)

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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.

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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.

**Note:** This article appeared in an earlier IntMath Newsletter. Many people asked me to publish it as a separate article.

1 Dec 2008 at 11:55 pm [Comment permalink]

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.

15 Jul 2009 at 4:08 pm [Comment permalink]

YOUR UNDERSTANDING AS WELL AS ABILITY TO EXPLAIN TRIGONOMETRY IS APPRECIABLE

19 Jul 2009 at 10:11 am [Comment permalink]

Johan:Thanks for the input.And thanks for your feedback, Deepti – glad you find it useful!

13 Jun 2011 at 10:53 pm [Comment permalink]

Fabulous article! Thanks.

I’d like to emphasize the difference between what needs memorizing and what needs understanding. Students need to memorize the 3 basic definitions (sine, cosine, and tangent). But they need to understand when the signs are positive and negative by drawing the angle, and looking at x and y coordinates.

14 Jun 2011 at 10:29 am [Comment permalink]

Thanks, Sue. Yes – those who are reluctant to draw tend to have more problems with trigonometry!

1 Jul 2011 at 6:45 pm [Comment permalink]

Hi! I like because u really assist me in different aspect of mathematics. Thank you so much

2 Jul 2011 at 5:39 am [Comment permalink]

@Asus1: You are welcome!

11 Jul 2011 at 11:51 am [Comment permalink]

Thanks to you, many gain good ideas and knowledge through your newsletter.

Your description about trig is really easier and very useful.

I have one doubt not yet being cleared after many references.

What is the difference between continuity and uniform continuity?

Without knowing this concept clearly, i find it difficult to learn real analysis paper.

11 Jul 2011 at 4:42 pm [Comment permalink]

@Suvinthra: This might help: WISC on continuity and probably the easier one to follow: MathCS.

18 Jul 2011 at 11:41 am [Comment permalink]

Thanks a lot……

And I hope now I would be able to get through the ideas of Real Analysis clearly..

This newsletter has proved to me as a boon for mathematics learners…

30 Jan 2012 at 8:30 pm [Comment permalink]

Defining sine and cosine with right triangles is highly insufficient and does not give a single clue how they are computed. It is better to define them as power series: (sum(a,b) means sum from a to b of the given expression, summation sign)

sin x = sum(i=0,infinity)(-1)^i x^(2i+1) / (2i+1)!

cos x = sum(i=0,infinity)(-1)^i x^(2i) / (2i)!

With these definitions, you can prove i.e sin’x = cos x:

sin’x = sum(i=0,infinity)(-1)^i (2i+1) x^(2i) / (2i+1)!

= sum(i=0,infinity)(-1)^i (2i+1) x^(2i) / (2i)!.(2i+1)

= sum(i=0,infinity)(-1)^i x^(2i) / (2i)!

= cos x.

1 Jul 2013 at 12:20 am [Comment permalink]

good newsletter. Helps people who are learning about trigonometry