# Circular Functions of Real Numbers

## Key Questions

• Her are the graphs of the six trigonometric functions.

$y = \sin x$

$y = \cos x$

$y = \tan x$

$y = \cot x$

$y = \sec x$

$y = \csc x$

I hope that this was helpful.

• The sine and cosine of an angle are both circular functions, and they are the fundamental circular functions. Other circular functions can all be derived from the sine and cosine of an angle.

The circular functions are named so because after a certain period (usually $2 \pi$) the functions' values will repeat themselves: $\sin \left(x\right) = \sin \left(x + 2 \pi\right)$; in other words, they "go in a circle ". Additionally, constructing a right-angled triangle within a unit circle will give the values of the sine and cosine (among others). This triangle (usually) has a hypotenuse of length 1, extending from (0,0) to the circumference of the circle; its other two legs are one of the axes, and the line between the axis and the point where the hypotenuse meets the circle.

Every circular function can be derived from the sine and cosine. Some easy and well-known ones:
$\sin \left(x\right) = \sin \left(x\right)$
$\cos \left(x\right) = \cos \left(x\right)$
$\tan \left(x\right) = \sin \frac{x}{\cos} \left(x\right)$
The reciprocal functions:
$\sec \left(x\right) = \frac{1}{\cos} \left(x\right)$
$\csc \left(x\right) = \frac{1}{\sin} \left(x\right)$ - note that this can also be written as csec(x) or cosec(x)
$\cot \left(x\right) = \frac{1}{\tan} \left(x\right)$
Some more obscure ones:
$e x \sec \left(x\right) = \sec \left(x\right) - 1 = \frac{1}{\cos} \left(x\right) - 1$
$e x \csc \left(x\right) = \csc \left(x\right) - 1 = \frac{1}{\sin} \left(x\right) - 1$
Some more archaic ones include versin(x), vercos(x), coversin(x) and covercos(x). If you wish, you can research these yourself; they are rarely used today.