‘Trigonometry’ derives from the Greek, meaning literally ‘triangle measuring’.
Because ‘trigonometry’ is such a long word, people often informally abbreviate it as ‘trig’.
The tools developed in trigonometry complement and expand those developed in
geometry.
For example, geometry's ‘ASA’ (anglesideangle) congruence theorem states that a unique triangle is formed by two angles and the included side. But what are the lengths (measures) of those remaining two sides? Trigonometry gives the answer.
Trigonometry is used heavily in
calculus (e.g., related rate problems and integration techniques),
which is why
it comprises a large part of any precalculus course.
Trigonometry is used in astronomy, navigation, architecture and landsurveying.
The trigonometric functions are invaluable in studying any periodic (repetitive) phenomena.
Love your cell phone and music CDs? Signal processing, noise cancellation, and music coding
all use trigonometry.
There are two basic approaches to trigonometry:
These two approaches are ‘introduced via sketches’ below, and discussed thoroughly
in the next two sections.
You must know both approaches, because different problems favor one over the other.
Although at first glance the two approaches might seem totally different,
they are
actually beautifully intertwined:
take one approach, apply a smidgeon of
similar triangle theory, and the other pops out!
Both the ‘right triangle’ and ‘unit circle’ approaches have the same goal—to define the primary two trigonometric functions.
These two functions are called sine and cosine:
formal (long) names:  sine  cosine 
short names used in function notation:  sin  cos 
pronunciations of both long and short names:  SINE (long i) (sounds like ‘sign’)  COsine (literally, ‘in company with the sine’) 
Recall that a function takes an input, and then uses this input to give a unique corresponding output.
For trigonometric functions, the input is the measure of an angle (for now), and the output is a real number.
For emphasis, let's repeat an earlier sentence with our new convention:
As you'll see below, the outputs from sine and cosine are extremely simple!
In the right triangle approach, they're just ratios of lengths of sides of a triangle.
In the unit circle approach, they're just coordinates of a point.
Oh, that such incredible power and utility can come from such simple beginnings!
The following information is common to both approaches:
Here are ‘in a nutshell’ looks at the two approaches.
For all the details, study the next two sections in this Precalculus course:
In the right triangle approach to trigonometry, the trigonometric functions are just ratios of lengths of sides in a right triangle! 
using the familiar $\,30^\circ$$60^\circ$$90^\circ\,$ triangle: $$ \begin{alignat}{2} \sin 30^\circ&\ \ =\ \ && \frac{s}{2s} &\ \ =\ \ && \frac 12\cr\cr \cos 30^\circ &\ \ =\ \ && \frac{\sqrt 3s}{2s} &\ \ =\ \ && \frac{\sqrt 3}2\cr\cr \sin 60^\circ &\ \ =\ \ && \frac{\sqrt 3s}{2s} &\ \ =\ \ && \frac{\sqrt 3}2\cr\cr \cos 60^\circ &\ \ =\ \ && \frac{s}{2s} &\ \ =\ \ && \frac 12 \end{alignat} $$ 
The circle above is centered at the origin and has radius equal to $\,1\,$. It is called the ‘unit circle’. In the unit circle approach to trigonometry, sine and cosine are just coordinates of a point on the unit circle! 
to get to the same terminal point: $30^\circ\,$ (positive angles are swept out counterclockwise; up) $330^\circ\,$ (negative angles are swept out clockwise; down) It doesn't matter how we get there! All that matters are the coordinates of the terminal point: $\displaystyle\cos 30^\circ = \cos(330^\circ) = \frac{\sqrt 3}{2}$ $\displaystyle\sin 30^\circ = \sin(330^\circ) = \frac{1}{2}$ 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
