# The Trigonometric Functions Secant, Cosecant and Cotangent

## 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

$\sec x = \frac{1}{\cos x}$

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

$\csc x = \frac{1}{\sin x}$

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

$\cot x = \frac{1}{\tan x}$

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 $$p(x) = \displaystyle \frac{1}{2 + \sin x}$$.

This is periodic with period $$2$$.

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

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

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