Trigonometric Functions as Projections

Similarity between sine and cosine functions

1. \(\sin(\theta) = \cos(\theta)\) when \(\theta=45^\circ\) (first quadrant) and again when \(\theta=225^\circ\) (third quadrant). Because sine and cosine are both repeat every 360 degrees, it is also true that \(\sin(\theta) = \cos(\theta)\) when \(\theta=-315^\circ\) degrees or \(\theta=405^\circ\) or \(\theta=-135^\circ\), etc. We can express this mathematically as \[\sin(\theta) = \cos(\theta)\text{ when }\theta=45 + n\times 180^\circ,\quad n\in \mathbb{Z}\]

   At these values of \(\theta\), \(\sin(\theta) = \cos(\theta) = 0.707\) (approximately). (The exact value is \(\sqrt 2\over 2\) which is irrational.)

2. \(\sin(\theta) = -\cos(\theta)\) when \(\theta=135^\circ\) (second quadrant) and again when\(\theta=315^\circ\) (fourth quadrant). Naturally this also repeats every 360 degrees, so \[\sin(\theta) = -\cos(\theta)\text{ when }\theta=135 + n\times 180^\circ,\quad n\in \mathbb{Z}\]

Challenge:

By comparing where sine and cosine reach their maximum and minimum and where they cross the axis you should be able to see that the cosine function takes the same values as the sine function, only 90 degrees earlier. This can be expressed mathematically as \[\cos(\theta) = \sin(\theta + 90^\circ)\] or alternatively \[\sin(\theta) = \sin(\theta - 90^\circ)\]

(There are other ways of writing this so if you got something a bit different from these, check it with your teacher.)

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