Inverse Circular Functions

Inverse circular functions or inverse trigonometric functions are the inverse functions of the trigonometric functions; sine, cosine, tangent, contangent, cosecant and secant.

Let the trigonometric functions such as sine, cosine, tangent, etc. be defined by $y=\sin x,\;y=\cos x,\;y=\tan x,\;\text{etc.}$

Let us express above functions as a set of ordered pairs $(x,y)$ of real numbers. Then, $f=\{(x,y):y=\sin x\},$ $g=\{(x,y):y=\cos x\},$ $h=\{(x,y):y=\tan x\},\;\;\text{etc.}$

On interchanging the roles of $x$ and $y$, we get, $F=\{(x,y):x=\sin y\},$ $G=\{(x,y):x=\cos y\},$ $H={(x,y):x=\tan y},\;\;\text{etc.}$

In each of these cases, each $x$ corresponds to more than one $y$. For instance, if $x=\frac{1}{2}$ in the $F$, then $y$ may be $30°$, $150°$, $360°+30°$, etc. so, none of them represents a function.

But, if we restrict the values of $y$ in these cases, then we can get only one value of $y$ corresponding to each value of $x$. Then, we get well defined functions. Let us explain this more briefly.

The sine function is a circular function defined by $y=\sin x$ i.e. $f=\{(x,y):y=\sin x\}$

In this function $f$, the domain is the set of real numbers, and its range is the set of real numbers between $-1$ and $1$ inclusive. $\therefore\text{Domain of}\; f=\{x:x\in R\}$ $\therefore\text{Range of}\; f=\{y:-1≤y≤1\}$

On interchanging the roles of $x$ and $y$, $F=\{(x,y):x=\sin y\}$

In this case, the domain of $F$ is the set of real numbers between $-1$ and $1$ inclusive and its range is the set of all real numbers. $\therefore\text{Domain of}\; F=\{x:-1≤x≤1\}$ $\therefore\text{Range of}\; F=\{y:y\in R\}$

Hence, several values of $y$ may correspond to one $x$.

This can be shown more clearly from the graphs of $y=\sin x$ and $x=\sin y$.

The graph of $y=\sin x$ winds along the x-axis. A horizontal line lying between $y=-1$ and $y=1$ intersects the graph of $y=\sin x$ at several points. This shows that there are many values of $x$ which corresponds to a certain value of $y$. One can also see that the length of the longest interval on the x-axis for which the horizontal line can intersect the curve $y=\sin x$ in at most one point is $π$. This interval is chosen to be an interval $-\frac{1}{2}π≤x≤\frac{1}{2}π$ (the heavy part of the graph). Moreover, this choice of domain gives a full range of values (i.e. $-1≤y≤1$) of the function defined by $y=\sin x$.

The graph of $x=\sin y$ is of the same form as $y=\sin x$ but winds along the y-axis. A vertical line lying between $x=-1$ and $x=1$ intersects the graph of $x=\sin y$ at more than one point. This shows that $x=\sin y$ does not define a function. But, if we restrict the values of $y$ to the solid part of the curve only so that $-\frac{1}{2}π≤y≤\frac{1}{2}π$, then each value of $x$ between $-1$ and $1$ inclusive corresponds to exactly one value of $y$.

We indicate this restriction in $x=\sin y$ by using the following notations: $x=\sin y$ $y=\text{Arc}\sin x$ $y=\sin^{-1}x\;\;\text{(read ‘inverse sine of x’)}$ and the value of $y$ so obtained is called the principal value (p.v.).

$x=\sin y$, $y=\text{Arc}\sin x$ and $y=\sin^{-1}x$ are three equivalent notations for the same thing. Also, it should be remembered that, $\sin^{-1}x\;\;\text{NEVER MEANS}\;\;\frac{1}{\sin x}$

Hence, we have the following definitions of the following inverse circular functions; the inverse sine, the inverse cosine and the inverse tangent.

The inverse sine function is defined by $\sin^{-1}y\;\;\text{or equivalently}\;\;x=\sin y$ provided that $-1≤x≤1$ and $-\frac{1}{2}π≤y≤\frac{1}{2}π$.

The inverse cosine function is defined by $\cos^{-1}y\;\;\text{or equivalently}\;\;x=\cos y$ provided that $-1≤x≤1$ and $0≤y≤π$.

The inverse tangent function is defined by $\tan^{-1}y\;\;\text{or equivalently}\;\;x=\tan y$ provided that $-\infty≤x≤\infty$ and $0≤y≤π$.