The Trigonometric Functions Secant, Cosecant and Cotangent

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Three More Trigonometric Functions

Three more trigonometric functions can be introduced; these are related to the sine, cosine and tangent functions.

The Secant Function

The secant function is normally written as \(\sec x\) and is defined by

\[\sec x = \frac{1}{\cos x}\]

Sec.jpeg

The function \(f(x) = \sec x\) goes off to \(\pm \infty\) at the values of \(x\) where \(\cos x = 0\). The function has a range between minus infinity and \(-1\) and also between \(1\) and infinity. The function is periodic with period \(2\pi\).


The Cosecant Function

The cosecant function is normally written as \(g(x) \mathrm{cosec}(x)\) or \(\csc(x)\) and is defined by

\[\csc x = \frac{1}{\sin x}\]

Cosec.jpeg

The function \(g(x) = \csc x\) goes to infinity periodically. Like the secant function it has a range between minus infinity and \(-1\) and also between \(1\) and infinity. It too has a period of \(2\pi\).

The Cotangent Function

The cotangent function is normally written as \(h(x) = \cot x\) and is defined by

\[\cot x = \frac{1}{\tan x}\]

Cot.jpeg


This function is a surjection, taking all real numbers as values. It is periodic with period \(\pi\).

\[\sec x = \frac{1}{\cos x}\]

\[\csc x = \frac{1}{\sin x}\]

\[\cot x = \frac{1}{\tan x}\]

Shift, stretches and periodicity

The secant and cosecant have the same period as the corresponding sine and cosine functions; the cotangent has the same period as the corresponding tangent function.

Functions such as \(\sec(ax+b)\) or \(\csc(ax+b)\) have period \(\displaystyle \frac{2\pi}{a}\). The function \(\cot(ax+b)\) has period \(\displaystyle \frac{\pi}{a}\).


\(\sec\) and \(\csc\) obey the same periodicity rules as \(\sin\) and \(\cos\)

\(\cot\) obeys the same periodicity rules as \(tan\)

Examples on Periodicity

Find the periods of the following functions. \(\sec(3t+5)\), \(\csc(x/2)\), \(\cot (5x)\)


Solution :-

Using the above guidelines, the periods are \(\displaystyle \frac{2\pi}{3}\), \(\displaystyle \frac{2\pi}{1/2} = 4\pi\) and\(\displaystyle \frac{\pi}{5}\).

Further Functions

Of course, there are many more ways that further trigonometric functions can be created / combined.

For example, consider the function \(p(x) = \displaystyle \frac{1}{2 + \sin x}\).

Modified.jpeg

This is periodic with period \(2\).

Sometimes a function can involve a trigonometric function and anoter function. See below for \(r(x) = \cos x \times e^{-x/4}\)

Cosexp.jpeg

This function is not periodic as the exponential causes the oscillations to decrease.

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