Trig Identities #3 Periodicity

tr2

You can hopefully see from the diagram above, that if we add 2\pi to our
angle \theta , we would come ‘full circle’ (ba-dum tish) and be back where we started from (think of it as adding 360 degrees if it makes you happier).

Hence:

(\cos(\theta \pm 2\pi), \sin(\theta \pm 2\pi)) = (\cos(\theta), sin(\theta))

\cos(\theta) = \cos(\theta \pm 2\pi)

\sin(\theta) = \sin(\theta \pm 2\pi)

And since csc and sec are basically just utilisations of sin and cos, then they have the same property:

\csc(\theta) = \csc(\theta \pm 2\pi)

\sec(\theta) = \sec(\theta \pm 2\pi)

Now  \tan \theta has a periodic cycle half the size of \sin \theta , which you can perhaps visualise easily by studying their respective plots side by side:

trig5

and so :

\tan(\theta) = \tan(\theta \pm \pi)

\cot(\theta) = \cot(\theta \pm \pi)

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