We know that there are two "special" right triangles that continually appear throughout the study of mathematics: the 30º-60º-90º triangle and the 45º-45º-90º triangle. The "special" nature of these triangles is their ability to yield exact answers instead of decimal approximations when dealing with trigonometric functions.

If you remember the labeled triangles below, you will be able to quickly express the exact value of any of the six trigonometry functions for angles of 30º, 45º, and 60º. These angles are also commonly used as reference angles in other quadrants.

45picmini     30picmini

These triangles will be used continually throughout your studies of mathematics and are well worth your time and study. 



The most powerful method of working with these triangles is to work directly with the triangles themselves.  By remembering the "relationships" in these triangles, you will have the power to apply any of the six trigonometric functions in any situation.

Some people find charts helpful for remembering the trigonometric values of 30º, 45º, and 60º.  Unfortunately, memorizing charts of information often gives you a limited understanding of the concepts at hand.  For example, if a chart is limited to only a few of the trigonometric functions, such as the charts seen below, your "power" to solve a multitude of problems is also limited.

  30º 45º 60º
sin sp1 4 6
cos 7 4 sp1
This chart shows the values with
rationalized denominators.
  30º 45º 60º
sin 8 9 11
cos 12 13 3
This chart shows a "trick" for remembering the values. 
Notice the row for sin shows 1, 2, 3 under the radicals,
while the row for cos shows 3, 2, 1 under the radicals.

Refresh your memory on working with 30º-60º-90º triangles and the 45º-45º-90º triangles,
if needed, and let's get started using these triangles in trigonometry.


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