Trigonometric Functions as Projections

Similarity between sine and cosine functions

I'm sure you have noticed a strong similarity between sine and cosine functions. This is hardly surprising since the definition of sine is very similar to that of cosine. Use the applet below to explore this as you plot sine (in blue) and cosine (in green) on the same axes. Then use the interactive to help with the exercises. (The applet shows three full cycles to make it easier to see the similarity.)

Exercises

1. At what angles does \(\sin(\theta) = \cos(\theta)\)? What is the value of \(\sin(\theta)\) and \(\cos(\theta)\) at these points?
2. At what angles does \(\sin(\theta) = - \cos(\theta)\)?

Challenge:

Look back on the earlier exercises you did for sine and cosine as well as at the graph generated by the applet above. Can you express mathematically the relationship between \(\sin(\theta)\) and \(\cos(\theta)\)?

Check your answers