Six Basic Trigonometric Functions

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In this section, we’ll briefly recapitulate the fundamentals of trigonometry, assuming that you’re already familiar with the material presented.



\((a)\;\;\; f (x) = sin\; x\)



\((b)\;\;\; f (x) = cos\; x\)



\((c)\;\;\; f (x) = tan \;x\)

\((d)\;\;\; f (x) = cot\; x\)

\((e)\;\;\; f (x) = sec\; x\)


\((f)\;\;\; f (x) = cosec\; x\)

It is important to be familiar with these six basic functions and their properties for further study of trigonometry.


Listed below are basic relations satisfied by these functions:
(A)

  • \({\sin ^2}x + {\cos ^2}x = 1\)
  • \(1 + {\tan ^2}x = {\sec ^2}x\)
  • \(1 + {\cot ^2}x = {\text{cose}}{{\text{c}}^2}x\)

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(B)

  • \(\sin \left( { - x} \right) = - \sin x\)
  • \(\cos \left( { - x} \right) = \cos x\)
  • \(\tan \left( { - x} \right) = - \tan x\)
  • \(\cot \left( { - x} \right) = - \cot x\)
  • \(\sec \left( { - x} \right) = \sec x\)
  • \({\text{cosec}}\left( { - x} \right) = - {\text{cosec}}\;x\)

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(C)

  • \(\begin{align}\sin \left( {\frac{\pi }{2} \mp x} \right) = {\text{cos}}\;x\end{align}\)
  • \(\sin \left( {\pi - x} \right) = \sin \;x\)
  • \(\sin \left( {\pi + x} \right) = - \sin \;x\)
  • \(\begin{align}\cos \left( {\frac{\pi }{2} \mp x} \right) = \pm \sin \;x\end{align}\)
  • \(\cos \left( {\pi - x} \right) = - \cos \;x\)
  • \(\cos \left( {\pi + x} \right) = - \cos \;x\)
  • \(\begin{align}\tan \left( {\frac{\pi }{2} \mp x} \right) = \pm \cot \;x\end{align}\)
  • \(\tan \left( {\pi \pm x} \right) = \pm \tan \;x\)

The properties in (A) follow from the definitions of these functions, while those in (B) and (C) can be inferred from the graphs of these functions.

 

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