Why are you forcing me to learn this stuff?
Trigonometry was originally all about the relationships that exist between the sides of a right triangle with respect to a given angle, and has since become an indispensable tool among architects, surveyors, astronomers, navigators, and others. This is especially true when it comes to spherical trigonometry, which is concerned specifically with the study of triangles on the surface of a sphere such as the Earth.
We can use trigonometry to determine distances and heights that cannot easily be measured directly, such as the height of a mountain too tall to climb, the altitude of a plane flying high overhead, or the width of a river too deep to ford. Astronomers also use it to calculate distances to objects in space, and there exist many other much more complex applications as well.
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Most importantly for you, knowledge of trigonometry will be required in order to understand calculus in the future. (Calculus is used to calculate the area or volume of shapes or figures that have curves which change at constantly varying degrees, or to calculate a moving object’s varying rate of speed at a given point along its path based on the moment in time it was there.)
THE FUNCTION OF FUNCTIONS
As previously stated, trigonometry was originally centered on the relationships existing between the sides of a right triangle with respect to a given angle.
You may recall from Algebra I and II that the mathematical concept which, at its very core, is all about relationships is that of a function.
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Of course: A function is a relationship from a set A to a set B in which each element in set A (known as the input) is assigned one and only one element from set B (known as the output) as dictated by some “rule of correspon-dence” or formula.
In the case of a right triangle, there exists six basic functions—six fundamental relationships between its three sides. These relationships are dealt with so often that they have each been given their own special name: sine, cosine, tangent, cotangent, secant and cosecant.
One of the most difficult tasks for some math students is that of memorizing the six trigonometric functions. No matter how much they study, the concepts are just too abstract for them, and unless they review the functions on an almost daily basis, the relationships are always once again forgotten.
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Some textbooks try to make this challenge easier by encouraging students to rely on such mnemonic nonsense as camp “Soh Cah Toa” and whatnot, but for some, that too is simply more gobbledygook. (It just sounds like some relative of Sacajawea or something!)
However, to introduce the trigonometric functions in a logical, easy to remember manner, this textbook will explain the concepts geometrically rather than algebraically, which only makes sense. For when it comes to studying advanced mathematics, one finds that the best approach is almost always to analyze the topic in the same way it was developed by its originators. So, let’s begin doing that right now…
TRIGONOMETRY
TRIGONOMETRY
THE UNIT CIRCLE
The ideas discussed in this section are foundational to developing a clear understanding of the trigonometric functions, so it is critical that you is your attention—like a laser beam—as each idea is introduced.
When it comes right down to it, a trigonometric function is simply an output value obtained by dividing one number by another. In just a few moments it’s going to be extremely important that you don’t lose track of this fact: When all is said and done, each of the six trigonometric functions is simply a number (or value).
Of course, generally speaking, a function can be defined as a relationship from a set A to a set B in which some formula is used to assign to each element in set A (known as the input), exactly one element from set B (known as the output).
