Many subjects in science require you to study angles and their relationship to the sides of triangles associated with them. An example is the concept of apparent size which plays an important roll in the study of optics and of astronomy. In geography and navigation, you will use the measurement of angles to determine longitudes and latitudes. These relationships also play an important role when you study rotational dynamics in physics. The study of angles and their interrelationships forms a branch of mathematics called Trigonometry.
There are six basic trigonometric functions which detail these relationships. They are; Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent. In LiveMath, as in most math books, they are abbreviated as follows; sin, cos, tan, csc, sec, cot. Note that LiveMath uses all lower case letters, unlike some other math programs.
For most of your study of calculus you will use radian measurement
for angles. Radian measurement of a central angle is the number
of radius units long the central angle's arc is. You know that
the radius of a circle is related to the circumference by the
formula: 
therefore the
circumference would be the length of the radius times 2 * pi.
The unit circle, whose radius is equal to 1 (one), is used to
study these functions. Therefore, the circumference is equal to
2pi.
LiveMath's Trig functions are in radians, as you will find in most computer programs. When given problems stated in degree measurement, you will have to convert them to radians. To do this in LiveMath you use the PreDefined name Degrees (represented with the degree symbol º). To convert from radians to degrees you will have to set up a Transformation rule (see the User's Guide, page 122). For reference the conversion formulas are shown below.
When a problem has angles and distances as part of its givens, it is useful for you to look at a right triangle for help finding its solution. You can define the trigonometric functions using the ratios of the sides of a right triangle in relation to one of its angles. In particular, this angle is an acute angle, which is an angle between 0º and 90º. Below, the symbol alpha is used to denote the angle. As a review, the following diagram highlights the three main functions, sin, cos, and tan.
This diagram only shows the three initial formulas. Another three; Cosecant, Secant, and Cotangent, are the reciprocals of the three shown, respectively.
In most of your study of calculus, you will be looking at Trigonometric functions as they are defined by the unit circle (see the next section).
There is one formula from this triangle group that has particular importance in calculus. This is the Tangent. You will often see mention of "tangent to the curve". Keep this simple Triangle Trig formula in mind as you progress in your study.
Many natural phenomenon are of a periodic nature. For example, a mass attached to a vibrating spring will repeat a similar motion over and over again. The rising and falling of the earth's tides also demonstrates a periodic motion. The oscillation of electric current is yet another example of this type of occurrence.
The "Unit" circle, and the trigonometry associated with it, is used to study these periodic phenomenon. This special circle has the following formula.
This formula states that the radius is equal to 1 (one). It
is easy to work with because, as the diagram below shows,
and 
are
simply the y and the x coordinates of the point
where the terminal ray 
of 
, intersects the circle. The radius being
equal to 1 makes for this simple relationship. The point is defined
as; ,
The remaining four Trig functions are defined as quotients and reciprocals of these two fundamental functions.
Since and are the
x and y coordinates, respectively, of a point on
the unit circle, you can develop the remaining functions using
the same methods shown in the previous section.
As you study trigonometry you will discover that both the Sine and Cosine functions produce repeating values for an infinite number of inputs (angles) according to the following two identities. Note
This means that as a point travels counterclockwise around the circumference of a Unit Circle 2 times, from any original point, the Sine or Cosine of the resulting angle will be the same. For this reason, these functions are said to be periodic with a period of 2pi. This becomes all the more evident when you graph these two functions.
This is informative, but a better way of looking at the relationship between the two functions would be to plot both on the same graph.
This method of adding a line plot to a Graph Theory does the trick, but has drawbacks. You can change the cosine function from the outside of the Theory and the result would be that the black line would change to match the new function you entered. On the other hand, to change the sine function would require that you have the graph details open. This non dynamic construction, however, may be the way you would prefer creating the Theory. For example, you may want to hide the function from viewers of your Web page or a teacher may want to hide a function from students taking a test.
You used the Unit Circle and triangle trig to determine that the Tangent of an angle is the Sine divided by the Cosine. Many other "Identities" are available to you by using a combination of Geometry and Algebra. These Identities are used often in calculus to manipulate functions into forms that are easier for you to work with.
The formula for a circle with a radius of one is:
Since 
are the x
and y coordinates of a point on this circle, you can state
the following identity;
Using simple algebra:
Many others are derived using the Pythagorean Theorem.
It is not the intent of this book to go into detail on this subject, but it is important to know that LiveMath allows you to have, at your fingertips, all of the standard identities, or any other you may wish to have, through the use of Transformation Rules. LiveMath provides many of these identities in the notebook called; "New Notebook" and a more complete list in the notebook called "Trig Functions", found on your program disks/CD.
The following dialog will open asking you to choose one of three transformations. Choose the middle one by clicking directly on top of it.
With the advent of the scientific calculator and personal computers, the process of finding the values of Trig Functions has become as easy as pushing a couple of buttons or inputting a value to a spreadsheet cell. This does not, however, relieve the curious from the quest to understand where these answers come from. Later in your study of mathematics, you will likely study Infinite Series, where you will learn how to derive these values.
LiveMath has built-in functions that will easily give you values.
LiveMath Internet: Sine Values
The following will demonstrate how to use LiveMath's graphing ability to determine the same value.
The solution is the root of the equation. The methodology behind this exercise is similar to that used in the inequalities example. You assign a variable (x in this case) and equate it to the value you want to find.
By making it a graphable function and finding the root, you
"solve for x". You do this by shifting everything
to the right side of the equal sign which creates a new equation:
. 
Substituting y for the
0 gives you an equation LiveMath can graph. Where y
= 0 is where x = sin(2).
LiveMath provides you a very useful tool that demonstrates the periodic nature of the sine function. By adding a constant to the function variable, a phase-shift takes place. When you animate this "parameter" a very interesting sequence appears.
LiveMath Internet: Sine Animation
LiveMath right-click or option-click to download
Finally, select a and choose Graph->Animation->Start. LiveMath automatically animates the plot for you. Each frame changes the argument for the sine function by the value of a incremented by the value of pi/5 from 0 to 2, thereby causing a shift in the plot.
You may want to experiment by changing the value given to a before the animation to see how the phase shift works.
