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Right Triangle Trigonometry

The basis of trigonometry is found in the study of right triangles.  These are triangles that contain a right angle.  A right triangle has six measurements that are of interest:  the measures of the three angles, labeled A, B, and C in the picture, and the lengths of the three sides, shown as a, b, and c.

In the right triangle below, note that we have used lower case letters “a,” “b,” and “c” to denote the lengths of the sides opposite of angles A, B, and C, respectively.  The side labeled “c” has a special name because it is the side opposite the right angle C.  It is called the hypotenuse of the right triangle.  If we focus on angle B, the side labeled b is called the side opposite angle B, while the side labeled “a” is called the side adjacent to angle B.  Similarly, side a is the side opposite angle A, while side b is the side adjacent to angle A. 


 

When we focus on one of the acute angles  (let’s call it , a Greek letter pronounced “theta”) of a right triangle, there are six possible ratios of side lengths in the right triangle; these ratios define the trigonometric functions of any acute angle , as given in the chart below.

 

 

These definitions allow us to set bounds on the values of the trigonometric functions of . Since the hypotenuse is always the longest side of a right triangle, then:

,   for

Note:  we use two different kinds of angle measures, namely degrees and radians.  When we measure angles in degrees, the symbol “°” must be used, while when working in radians, we don’t have to write the word “radians.”  Measures given without units are assumed to be radian measures.  The angle measure of radians shown above is equal to 90°, and is, of course, a right angle measure.

Notice also that:
                                                     
                                                     
                                                     

We sometimes call these reciprocal functions, or reciprocal identities.

Also note that the other acute angle of the triangle has measure .  Recall that the sum of the three angles in any triangle is 180°, or  radians.  The right angle measures , so the sum of the two acute angles must also equal  .  Then the side opposite one acute angle  in a right triangle is the side adjacent to  , the other acute angle in the triangle (and vice versa). Therefore  and  .  We may of course use these facts to establish other identities.


Two Special Triangles


         

Recall from geometry that there are two right triangles with particularly “nice” side length ratios. The (or 30°—60°—90°) right triangle has sides in the ratio , while the (or 45—45—90, in degrees) right triangle has sides in the ratio .  With these facts, we can find the trigonometric function values for “special” angles.   Using the definitions given for the six trigonometric function ratios, we can find that, for example, .