In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Nataly Arias. Her topic, from Precalculus: computing trigonometric functions.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Trigonometry does not only relate to mathematics, trigonometry is also used in real life. Many people don’t know that trigonometry is involved in video games. In game development, there are many situations where you will need to use trig functions. Video games are full of triangles. For example in order to calculate the direction the player is heading you will form a triangle and use sine, cosine, or tangent to solve. The trig function used depends on the values given. For example if the opposite and adjacent values are given (the xSpeed and ySpeed), the function you will need to calculate the direction of the player is tangent. This is represented by the equation Tan( Dir ) = xSpeed /ySpeed. Again, by applying the inverted function of tan to both sides of the equal sign, we get an equation that will return the player’s direction. In a spaceship game you will need to use trigonometric functions to have one ship shoot a laser in the direction of the other ship, play a warning sound effect if an enemy ship is getting too close, or have one ship start moving in the direction of another ship to chase. Trig is used in several situations in video games some more examples include calculating a new trajectory after a collision between two objects such as billiard balls, rotating a spaceship or other vehicle, properly handling the trajectory of projectiles shot from a rotated weapon, and determining if a collision between two objects is happening.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

The “unit circle” is a circle with a radius of 1 that is centered at the origin in the Cartesian coordinate system in the Euclidean plane. Because the radius is 1 we can directly measure sine, cosine, and tangent. The unit circle has made parts of mathematics easier and neater. The concepts of the unit circle go far back into the past. Not only do we use and see circles in mathematics we also can see circles in art form. We can also use trigonometric functions to determine the best position to view a painting hanging on an art gallery wall. For example you can determine the angle between a person’s eye and the top and base of the painting when a person is standing 1m away, 2 m away, 3 m away and so on. By comparing your data you can estimate the best position for a person to stand in front of the painting. Also using trig functions and your handy calculator you can develop a formula that describes the relationship between the distance away from the painting and the angle that exists between the person’s eye and the top and bottom of the painting.

How have different cultures throughout time used this topic in their society?

Today the unit circle is used as a helpful tool to help calculate trig functions. Trig functions are taught in trigonometry, pre-calculus and are frequently used in advanced math classes. Many people don’t realize that not only are trig functions learned and used in school but throughout time several cultures have used trig functions in their society. The main application of trigonometry in past cultures was in astronomy. In 1900 BC the Babylonians kept details of stars, the motion of planets, and solar eclipses by using angular distance measured on the celestial sphere. In 1680-1620 BC the Egyptians used ancient forms of trigonometry for building pyramids. The idea of dividing a circle into 360 equal pieces goes back to the sexagesimal counting system of the ancient Sumerians. Early astronomical calculations wedded the sexagesimal system to circles and the rest is history. Today in trigonometry the unit circle has a radius of 1 unlike the Greek, Indian, Arabic, and early Europeans who used a circle of some other convenient radius. In today’s society trigonometry is everywhere. The mathematics used behind trigonometry is the same mathematics that allows us to store sound waves digitally onto a CD. We use it without even knowing it. When we plug something into the wall there is trigonometry involved. The sine and cosine wave are the waves that are running through the electrical circuit known as alternating current.

**References**

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http://www.math.ucdenver.edu/~jloats/Student%20pdfs/40_Trigonometry_Trenkamp.pdf

http://www.math.dartmouth.edu/~matc/math5.geometry/unit9/unit9.html

http://en.wikipedia.org/wiki/Trigonometric_functions

http://aleph0.clarku.edu/~djoyce/ma105/trighist.html

http://www.slideshare.net/mgeis784/building-the-unit-circle

http://www.softlion.nl/download/article/Trigonometry.pdf

http://www.raywenderlich.com/35866/trigonometry-for-game-programming-part-1

http://stackoverflow.com/questions/3946892/trigonometry-and-game-development