You have studied polygons in the past.  Pentagons (5-sided), hexagons (6-sided), and octagons (8-sided) are commonly used terms in mathematics.  Trigon is not so common, but is simply another name for triangle and "metry" means to measure.  Thus Trigonometry is the study of "measuring triangles".

Trigonometry functions allow you to measure/calculate the lengths of sides and angles in a triangle.  For this unit we will concentrate on the right triangle.

In trigonomety, we call the three sides of the right triangle adjacent, opposite, and hypotenuse instead of  a, b, and c.  The sides of the acute <A are the adjacent side and the hypotenuse.  The opposite side completes the right triangle.  The opposite and adjacent sides, always form the right angle.

To remember the trigonometry functions, use the memory aid:

Soh      Cah      Toa

<A HREF="trig!.wav">Play with stand-alone Real Player</A>

The first letter of each aid represents the function.  The last two letters indicates the ratio of sides needed to calculate the function.

Soh   -->   Sine = opposite/hypotenuse

Cah   -->   Cosine = adjacent/hypotenuse

Toa   -->   Tangent = opposite/adjacent

We can use trigonometry functions to determine:

• one acute angle if you know two sides
• one of the sides if you know an acute angle and one side

Waves

There is beauty and pattern in trigonometry.  Click on "Start" and watch the wave generated for each trigonometyr function.  What pattern do you observe for each function?  Which two functions produce similar waves?

The sine wave starts at 0, moves up to 1,  down to  -1 and then back to 1.  The pattern continues with the wave moving between 1 and -1.

The cosine wave starts at 1, moves down to 0,  continues down to  -1 and then back to 1.  The pattern continues with the wave moving between 1 and -1.  The wave generated looks very much like the sine wave.

The tangent wave starts at zero, increases to infinitely positive, then moves down to  infinitely negative. The pattern continues with the wave moving between infinitely positive and infinitely negative.

source author:  J. David Eisenberg

Sine Cosine Tangent

What is the pattern of Sine, Cosine, and Tangent as you increase the angle from 0 to 90^o?  Use the radio buttons to select Sine, Cosine, or Tangent.   Move the radius of the circle (hypotenuse of the right triangle) from 0 to 90^o and observe what happens to the ratio!  Check the text and animated Gifs below to make sure you understand the pattern for each trigonometry ratio..

Java Applet compliments of  Walter Fendt

Trigonometry Reference Angles

The last applet helped you explore reference angles. These will help you estimate the angles in the triangle in the next section. Your estimation skills will improve as you learn the trigonometry ratios for other angles. The memory tips from the previous chart have been included:

<A HREF="trig!.wav">Play with stand-alone Real Player</A>

Six Trigonometry Functions

At what angle are the sine and cosine value the same?

After manipulating this applet you should see the sine and cosine ratios are the same (0.707) at 45^o.

from:  Six Trigonometry Functions

Visual Trigonometry

Trigonometry Circle - compare sine and cosine

Trigonometry Functions

Cogtech Trigonometry Function Explorer

Trigonometry ratios are calculated from an acute angle or any two sides of a right triangle ABC.  The buttons below show the angle and/or sides needed to calculate the trigonometry function you will need to calculate.  Click a button to see other examples of how to calculate trigonometry function from the angle or sides selected.