Essential Math - Trig

Trig Essentials

Angles

Most of us are familiar with the idea of an angle. In the diagram below, as one moves point B, one can see the angle changing. That's pretty cool, but in order to be useful, we need to have some way of measuring the angle.

Most people are familiar with measuring angles in degrees.  We simply divide a circle into 360 equal parts and define each part to be one degree.  We use this system in our daily lives. For example, in reading a compass or in saying that one has turned around 180 degrees.

There is nothing wrong with dividing a circle into 360 equal parts, but it is entirely arbitrary.  We could just as easily have divided the circle into 10 or 100 equal parts. But there is another, less arbitrary means of dividing a circle into parts.  That method uses radians as the unit of angular measurement.  Don't let the name scare you.  We're still just dividing a circle into smaller parts.  The only difference with radians is that they don't divide a circle into an equal number of parts (this might seem like a bad idea to you, but as will be shown shortly, the benefits are worth this small disadvantage).

In the figure above, the circle is divided into six radians.  Notice that six radians do not completely fill the circle.  In fact there are 2π radians in a circle. And since π = 3.1416, two times π will be 6.2832... a little more than six radians as shown in the figure.

So what would be the reason to divide a circle into anything other than equal parts?  The reason is that a radian is not an arbitrary division of the circle.  Instead it is directly related to the size of the circle.  In fact, it is defined by the radius of the circle, hence the name radian.  A radian is defined to be the angle between two points on a circle when the arc separating those two points is equal to the radius of the circle.

In the above figure, R is the radius of the circle, and S is the arc between the points 'a' and 'b' on the circle.  When the arc, S, has the same length as the radius, R, then the angle between 'a' and 'b' is one radian.  Now we begin to get to the nice part.  With this method of measuring angles, the measure of any angle is simply the ratio of the arc length to the radius of the circle.  In equation form that would be:

 

So if S is twice the length of the radius, then the angle is two radians.  Now it is more common that we can measure the angle θ, but don't know what R is, so the above equation can be rearranged as:

You'll see in a moment why this is useful.

 

 

Small Angle Approximation

Click and drag point B.  Notice that the arc length BDE and the chord length BE approach the same length as the angle BAE becomes smaller.

Sin Cos Tan of a Right Triangle

For any right triangle, the definition of the sine, cosine, and tangent of an angle in the triangle is simply the ratio of two sides of the triangle as defined below.

 

I have provide what I think are some of the most essential things from trigonometry needed for understanding the material on this site; however, I highly recommend the following site for an excellent review of trigonometry.

Dave's short course in trigonometry

Dave's site is probably the most informative and enjoyable site I've seen on the subject.  In fact, most of the interactive diagrams here were created using Dave's incredible Geometry Applet.  Many thanks to Dave for making this available.