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Trigonometry

Six Basic Trigonometric Functions

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)

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