Trigonometry is a branch of geometry; one of the major fields in mathematics. The name literally means, “the measuring of triangles” (from Greek trigōnon “triangle” + metron “measure”). As a child in school, I had a hard time understanding why the teacher thought it was so important for us to understand triangles. When students in my class raised this question, the answer was typically that we should just be quiet and do our work without questioning the usefulness of learning trigonometry. But it is difficult for anyone to work on something that is meaningless to them, we generally do our best work when it means something to us, that is, when we see a purpose for our labours. So the first task a teacher of trigonometry must do is reveal to the person you are teaching why triangles are worth studying at all. As such this article is not meant as an in-depth discussion of trigonometry, merely explaining the usefulness of the theory and pointing out various ways one could make it interesting for children to want to learn.
Why Should I Care About Triangles?
This is the first question anyone teaching trigonometry needs to answer first before any lessons commence. If you as a teacher of others cannot answer this question adequately then you will struggle to find any motivation or interest from the person you are trying to teach. So when planning your lesson, consider the various reasons why anyone would want to learn trigonometry. In the publicly funded school environment the motivation seems to be passing a test so the teacher can tell how good you are (this is teaching narcissism, that your value as a person comes from the approval or your usefulness to/of other people), in turn the teacher wants you to learn it well so they can get a good report and maybe a pay rise or even just keep their job. Hence why teachers can be quite aggressive, “learn it because I told you too!” which is not professional behaviour.
Here are a few other reasons that I came up with, many more exist, keep in mind that different people will be impressed with different reasons for why they should be interested in trigonometry:
Trigonometry is useful in construction. If your pupil is interested in construction, surveying or landscaping then teaching then trigonometry will be an essential skill. Even if there are now many devices that will do most of the work for them, it is important that they at least understand how the devices works so they can cope when they do not have access to it or to check for any errors in the device’s operation.
Trigonometry is used to measure the distances of stars. How do we know that the nearest star system to Earth is Alpha Centauri? Through triangulation understanding how maps and navigation work.
Trigonometry can be used to find things that you are looking for. Is there an annoying radio signal interferring with your TV reception but you can not figure out what it is and where it is coming from? By using triangulation one can locate the source of radio waves, sound waves and many other invisible phenomena.
Trigonometry can be used to find out how tall something is. Just say that you have a tree that you want to cut down, but it is dangerously close to a building. You can easily work out the distance of the building along the ground, but how do you know the height of the tree? Trigonometry is the easiest solution.
Basic Facts About Triangles
Triangles are geometric shapes, this means they defined mathematically, unlike say the shape of tree, a stone or gnawed chewing gum, or in fact any real world object technically. Shapes like circles, squares and triangles only exist by definition, and are only true by that definition. The definition of a triangle is any shape with three angles.
Consider that a shape with three sides might have more angles if you bent it into a three dimensional shape, so the definition of a triangle as a three sided shape is incorrect. If you are a student you might want to test your teacher’s knowledge with this question and gauge their response to it.
Because a triangle is a geometric shape, it has predictable qualities that can be used to figure out missing information. Consider the equilateral triangle below. Because we know an equilateral triangle has three sides and angles of equal dimensions, all we need to know is the length of side ‘a’ and we know the lengths of sides ‘b’ and ‘c’. This also applies to the angles, knowing ‘A’ means that we automatically know the angles for ‘B’ and ‘C’. In philosophical terms these are called necessary truths, they need to be true because the definition of a triangle states that they are so. It is impossible to have a triangle with four angles as this is outside of the definition of a triangle, therefore four angled triangles do not exist. Triangles with two angles of equal measure are called isosceles and those with three different angles are called scalene.
Next we have a triangle with a right angle, this is the angle labeled ‘A’. A right angle has exactly 90 degrees. Because the sum of angles in a triangle is by definition 180 degrees we can know with certainty that the sum of angles B and C will equal 90 degrees as 180 minus 90 equals 90. This is the advantage of using geometric shapes, unlike natural shapes, their definition is not open to interpretation, in the case of a triangle, once the angles have been calculated then the ratio of the length of each side is established mathematically, which means it is known with certainty. Some important vocabulary: the longest side of a right angled triangle is called the hypotenuse. If the triangle is scalene, then the side adjoining to the smallest angle attached to the hypotenuse is called the adjacent side. The side opposite the smallest angle is merely called the opposite side. If the triangle is a right-angled isosceles then we already know the angles and the ratios of the lengths for the sides so naming them is not important nor practical.
Seeing Triangles Everywhere
Once one has communicated to a person the power of triangles to discover things we can not easily measure. Now is the time to teach children to look for triangles. This can easily be turned into a fun game. To determine the height of a building or a tree, just look for the shadow: the edge of the shadow to the top of the building or tree forms a triangle. The edge of the building or the trunk of the tree with the ground is your right angle. Using a compass or a measuring tool like a quadrant or a sextant (you can make your own as a craft activity in your maths class!) measure the angle from the edge of the shadow to the top of the building or tree.
Remember, every square and rectangle can be turned into a triangle, so a TV screen can be measured diagonally to work out its width and height if one other angle is known. For advanced kids, triangles can be broken down into more triangles!
But the triangles that impress me the most:
If one child in Melbourne has a friend in Sydney, the two of them can take a measurement of the angle of the moon at the same time, share their readings and using the known distance between Sydney and Melbourne create a massive triangle to work out how far away the moon is from them at that point in time. This same procedure was done with the sun (not safe for children to attempt because of possible eye injuries) centuries ago to measure the distance from the Earth to the sun. Using a measurement taken in summer and another in winter they were able to work out the average distance the Earth was from the sun in an orbit. This was called an astronomical unit. Using a line one astronomical unit long from one side of Earth’s orbit to the other astronomers were able to measure the distances of the stars! All of this, using trigonometry.
Contained within mathematics are powerful cognitive tools for making sense of the world. In this short introductory article on trigonometry I hope you are starting to appreciate that a lot more is actually going on than just learning to memorise a set of formulas. The abstract nature of triangles invites children to learn about philosophical concepts such as universal or necessary truths like the definition of numbers and geometric shapes. Logical thought processes are used to gain new knowledge from existing definitions of these ideas. Thus mathematics is a great subject to teach logic and to teach practical problem solving. Considering all of the amazing things one can do with mathematics it is a shame that maths classes are seldom hand-on practical exercises working with hands, tools, materials and venturing beyond the classroom. If you are a maths teacher, set yourself a challenge to make your subject more practical and educational for your pupils!
Some jokes about triangles to entertain your children with:
Trigonometry jokes are a sine of the times.
Fake tan: The major threat to trigonometry
Q What does trigonometry have in common with a beach?
A: Tan Gents
Trigonometry for farmers: swine and coswine.
When were trigonometry tables used? “B. C.”, Before Calculators.
The Planes Indians practiced polygamy, and one chief had three squaws. The first squaw lived in a teepee of elk hide, the second in a teepee made of buffalo hide, and the youngest in a teepee of hippopotamus hide. Then he slept with each wife on the eve of his great hunting trip. He was gone nine moons and when he returned, he went into the elk hide teepee and found that his wife had borne him a son. Likewise, in the buffalo hide teepee, that squaw, too, had borne him a son. So, imagine his surprise when he found twin baby boys in the hippopotamus hide teepee.
This just proves that … The squaw of the hippopotamus is equal to the sum of the squaws of the other two hides.
Post Last Updated: 10/8/2013
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