Step 1:  Construct an imaginary right angle triangle to prove.

              CONSTRUCTION:  The angles in a triangle add up to 180. Since one

              angle  in a right triangle measures 90, the remaining two angles must  

              add up to 180 minus 90 or 90.

              So, if we name one of the angles \theta, then the other must

              be 90 minus  \theta .

              This ensures that the two angles sum to 90. When two angles sum to 90,

              we call them complementary angles.



Step 2: See how the sine of one acute angle

            EXAMPLE: \sin \theta = \frac{opp}{hyp} = \frac{m}{l}..................(1)

Step 3: Find the \cos of another acute angle.

            EXAMPLE:   \cos (90 - \theta) = \frac{adj}{hyp} = \frac{m}{l}.....................(2)


Step 4: Compare equation (1) & (2)

          EXPLANATION: \sin \theta = \cos (90 - \theta) = \frac{m}{l}            

         Incredible! Both functions,  \sin(\theta) and  \cos(90 - \theta), give the exact same side ratio in a right triangle.

            And we're done! We've shown that  \sin(\theta) = \cos(90 - \theta).