Introduction
Once in a while a topic is presented and its usefulness is instantly apparent. You immediately ask yourself how you got through life thus far without knowing
about such an amazing concept.
I haven’t studied math since high school, but I still remember one fateful morning in geometry class that changed my life.
Background
As a youth back in school it wasn’t uncommon when presented with a new topic, to hear comments like “When will I ever use this in life?” I remember having one
of these moments in grade 10 math class, when we spent a solid five months studying parabolas and quadratic equations. In hindsight, the problem was that we weren’t given any context
for how these curves could be useful. The teacher taught that which the curriculum dictated, but any significance of these equations was lost in translation.
The Triangle
I think it was in grade 7 that we were presented with the Pythagorean theorem. With the knowledge of the lengths
of two sides of a right-angle triangle, the theorem explains how to determine the length of the third side.
How useful! With this knowledge you can safely position an extension ladder to clean out the eaves troughs, calculate the shortest distance between two points
on a map, or buy enough wallpaper to cover an angled wall knowing only the dimensions of the other walls in the room.
Note: Americans call these “right triangles”. Please forgive them.
It felt like I was missing out on the full potential of this handy little triangle though. I wasn’t sure what else it could do, but something was missing.
One morning in grade 13 geometry class, the missing pieces presented themselves. The angles! Of course! Sure, the Pythagorean Theorem explained the relationship
between the sides, but we had forgotten about the angles. In grade 7 we accepted that one of the angles is 90° and that all three angles total 180° but we left it at that.
Trigonometry
That fateful trigonometry lesson in grade 13 explained a lot: how to determine any angle in the triangle, knowing the length of two sides; how to determine the length
of any side, knowing the length of one side and one angle; and just for what purpose those mysterious tan, cos, and sin buttons on my calculator were.
Let’s review.
- All triangles have three sides.
- In a right-angle triangle, two of those sides meet at a right angle (90°).
- The third side, called the hypotenuse, completes the shape.
Here’s a triangle. The little square opposite the hypotenuse denotes that it is a right-angle triangle (if it’s not present, we can not assume the angle is 90°).
To make the rest of this lesson easy to understand, let’s draw a little guy next to one of the points (called vertices)
of the triangle. His name is Dieter. The usefulness of this simple act cannot be overstated, because now we can use relational terms to describe the two unnamed sides. The side
next to Dieter, the one adjacent to him is called… the adjacent side. The side opposite him is… you guessed it, the opposite side.
Now that the sides are named we can apply some handy formulae. Don’t worry; they’re as simple as Pythagoras’ famous formula. We’ll use the symbol θ (Greek
letter theta) to represent the angle closest to Dieter. We’ll abbreviate opposite and adjacent to OPP and ADJ.
sin θ = OPP / HYP
cos θ = ADJ / HYP
tan θ = OPP / ADJ
“Why aren’t you writing this down?” The sine, cosine, and tangent functions are called
trigonometric functions.
Dieter is at the bottom of a hill at sea-level. He’s going to climb to the top. What will his altitude be when he reaches the top?
Let’s say the angle θ is 25° and the hypotenuse is 100 metres. His final altitude is equal to the length of the opposite side.
sin θ = OPP / HYP
OPP = HYP · sin θ
OPP = 100 · sin(25°)
OPP ≈ 100 · 0.4226
OPP ≈ 42.26
Dieter’s altitude will be about 42.26m above sea level.
Tip: Make sure your calculator is set to degrees, not radians, when working with degrees.
We can also do the opposite, determine the angle using inverse trigonometric functions.
If Dieter walked 100m up the hill and at the top his altimeter reads 35m, what is the angle of incline?
sin θ = OPP / HYP
θ = sinˉ¹(OPP / HYP) = arcsin(OPP / HYP)
θ = arcsin(35 / 100)
θ = arcsin(0.35)
θ ≈ 20.4°
The angle of incline is about 20.4°. The inverse of the sine function is called the arcsine and is usually denoted by the negative exponent (don’t let it confuse you).
Circles
If you think about it, every point on the circumference of a circle can be expressed in terms of triangles.
Here’s a task I’ve had to do many times, like when I wrote the Shoe Label
preview for Mabel’s Labels. When placing the letter d in the illustration, what are the x and y coordinates if we need to rotate
the text 30° and the radius is 50?
sin θ = OPP / HYP
x = sin θ · r
x = sin(30) · 50
x = 0.5 · 50
x = 25
cos θ = ADJ / HYP
y = cos θ · r
y = cos(30) · 50
y ≈ 0.866 · 50
y ≈ 43
The letter d can be placed at coordinates (25,43) at an angle of 30°.
Demonstration
The demonstration application uses Silverlight to demonstrate the concepts of right-angle triangles and tangential functions to animate
an elastic collision between a moving ball and a fixed post.
There isn’t anything particularly interesting about the application, but you can see these concepts expressed in code. The animation doesn’t take advantage
of Silverlight’s animation features, so it may be less than smooth.
undefined
Conclusion
If you remember three simple formulae and learn to use three buttons on your calculator, you’ll be able to amaze the other 72%¹ of the population who can not.
Addendum
¹ When I originally wrote this article, I made up a statistic for the percentage of people that are able to use these formulae.
I became curious about the actual number, so I initiated a research project.
In my experiment I paid 547 people from around the world (using Amazon Mechanical Turk)
USD 0.01 to solve Dieter’s problem above. The answers were multiple choice, including six numeric answers in addition to “I have no idea how to solve this.”
and “I remember doing this in school, but I forget how to solve it.”. A penny isn’t much money, but they weren’t told
that they would get paid whether or not the solution was correct. They were given five minutes to solve the problem. For those
with a working knowledge, the problem could easily be solved in about thirty seconds. Some might have gone to Wikipedia
and figured it out; others might have just clicked anything to move on to the next HIT. The results were as follows:
- 8% “I have no idea”
- 40% “I remember doing this in school”
- 24% An incorrect answer or guess
- 28% The correct answer
Acknowledgements
These things take a long time to write. If you like it, please rate it.
The animated Pythagorean triplet is © 2007 GNOME Icon Artists and is used under the terms
of the Creative Commons Attribution-ShareAlike 3.0.
Dieter is © 2007 AmericanExplorer13 and is used under
the terms of the GNU GPL.
