Trigonometry Triangles

by Victor Chien

1. Place the names of the six trigonometric functions in the
order in which we learned them (sin x, cos x, tan x, cot x, sec x,
and csc x) at the vertices labeled A, B, C, D, E, and F,
respectively.

2. Shade or color in the triangle with vertices A, B, and the
center. Then shade in the triangle whose vertices are
at D, F, and the center, and the triangle whose vertices
are at C, E, and the center.

            1. Cofunction Relations.
                 The trig functions cosine, cotangent, and cosecant on
                 the right of the hexagon are cofunctions of sine, tangent,
                 and secant on the left, respectively.

            2. Reciprocal Identities.
                 The two trig functions on any diagonal are reciprocals
                 of each other. Write the six identities below:

                 ______________________          ______________________   

                 ______________________          ______________________   

                 ______________________          ______________________   


            3. Product Identities.
                  Along the outside edges of the hexagon, any trig
                  function equals the product of the functions on the
                  adjacent vertices. Write these six identities below:

                 sin u = (cos u) (tan u)          ______________________   

                 _______________________          ______________________   

                 _______________________          ______________________   


            4. Pythagorean Identities.
                  For each shaded triangle, the upper-left function
                  squared plus the upper-right function squared equals the
                  bottom function squared. Write these three identities below:

                 ______________________          

                 ______________________           

                 ______________________          



            5. Examples of how the hexagon can be used to solve trig
               problems:

               A. Given sin u = 3/5 and tan u = -3/4, find the values
                  of the other six trig functions.
                     STEP 1: Write the given values of the trig function
                             at the appropriate vertices for sine and
                             tangent.
                     STEP 2: Find csu u and cot u by reciprocal relations. 
                     STEP 3: Find cos u by product relation.
                     STEP 4: Find sec u by reciprocal relation.
                    Write down your answers for the four trig functions
                    above, and draw a triangle to verify your answers.
               B. Given cot u = -8/15 and sec u > 0, find the values of
                  all trigonometric functions.
                    STEP 1: Write -8/15 at the vertex corresponding to
                            cot u. Note u is a 4th quadrant angle.
                    STEP 2: Find tan u by reciprocal relation.
                    STEP 3: Find sec u from the shaded triangle.
                    STEP 4: Find cos u by the reciprocal relation.
                    STEP 5: Find sin u by product relation.
                    STEP 6: Find csc u by reciprocal relation.