Tuesday, January 16, 2024

18.7 - Properties of Inverse Trigonometric Functions

In the previous section, we saw some solved examples on the six inverse trigonometric functions. In this section, we will see the properties of inverse trigonometric functions.

First we will see how the inputs and outputs are to be denoted. It can be written in 3 steps:  

1. In the discussions so far in this chapter, we gave primary importance to the original trigonometric function.
• For example, the original trigonometric function was:
y = f(x) = sin x
    ♦ Input was denoted as ‘x’.
    ♦ Output was denoted as ‘y’.
2. From the original trigonometric function, we derived the inverse trigonometric function.
• For example:
x = f(y) = sin-1 y
    ♦ Input was denoted as ‘y’.
    ♦ Output was denoted as ‘x’.
3. We have learned the basic details about inverse trigonometric functions. We can now treat them independently. So from now on wards, the inverse trigonometric function will be our primary function.
    ♦ Input for the inverse trigonometric function will be denoted as ‘x’.
    ♦ Output for the inverse trigonometric function will be denoted as ‘y’.


Now we will see an interesting case. It can be written in 3 steps:
1. In the previous chapter, we saw composite functions. Let us recall the details:
(i) f: A→B and g: B→C are two functions.
• Then the composite function g(f(x)) connects the domain of f with the codomain of g.
• So we can write:
    ♦ Domain of g(f(x)) is A
    ♦ Codomain of g(f(x)) is C
(ii) Suppose that, g is the inverse of f.
• Then f is mapped from A to B and g is mapped from B to A.
• In such a situation, we can write:
    ♦ Domain of g(f(x)) is A
    ♦ Codomain of g(f(x)) is also A
This is because, g(f(x)) connects the domain of f with the codomain of g.
• Also, g(f(x)) is an identity function. Whatever value we give as input, the output will be that same value.
(Recall that, function of “the inverse of that function” is an identity function)
• Some examples can be seen here.
(iii) Similarly, we can write about f(g(x)):
    ♦ Domain of f(g(x)) is B .
    ♦ Codomain of f(g(x)) is also B.
• f(g(x)) is also an identity function. Whatever value we give as input, the output will be that same value.
2. Now we will apply the above information on sin function and it’s inverse.
(i) Let us write the domain and codomain:
    ♦ sin function is mapped from $\left[-{\frac{\pi}{2}, \frac{\pi}{2}} \right]$ to [-1,1]
    ♦ sin-1 function is mapped from [-1,1] to $\left[-{\frac{\pi}{2}, \frac{\pi}{2}} \right]$
(ii) sin(sin-1 x) will connect the domain of sin-1 with the codomain of sin.
• That means, sin(sin-1 x) is mapped from [-1,1] to [-1,1]
• In other words, sin(sin-1 x) is a function on [-1,1]
• Also, sin(sin-1 x) is an identity function. Whatever value we give as input, the output will be that same value.
(iii) sin-1(sin x) will connect the domain of sin with the codomain of sin-1.
• That means, sin-1(sin x) is mapped from $\left[-{\frac{\pi}{2}, \frac{\pi}{2}} \right]$ to $\left[-{\frac{\pi}{2}, \frac{\pi}{2}} \right]$
• In other words, sin-1(sin x) is a function on $\left[-{\frac{\pi}{2}, \frac{\pi}{2}} \right]$.
• Also, sin-1(sin x) is an identity function. Whatever value we give as input, the output will be that same value.
3. The above step 2 is applicable to all six trigonometric functions. So we can write 12 identity functions:

$\begin{array}{cc}{}    &{\text{(i)}}    &{\sin \left(\sin^{-1} (x) \right)}    &{\text{(vii)}}    &{\sin^{-1} \left(\sin(x) \right)}    &{}\\
{}    &{\text{(ii)}}    &{\cos \left(\cos^{-1} (x) \right)}    &{\text{(viii)}}    &{\cos^{-1} \left(\cos (x) \right)}    &{}\\
{}    &{\text{(iii)}}    &{\tan \left(\tan^{-1} (x) \right)}    &{\text{(ix)}}    &{\tan^{-1} \left(\tan (x) \right)}    &{}\\
{}    &{\text{(iv)}}    &{\csc \left(\csc^{-1} (x) \right)}    &{\text{(x)}}    &{\csc^{-1} \left(\csc (x) \right)}    &{}\\
{}    &{\text{(v)}}    &{\sec \left(\sec^{-1} (x) \right)}    &{\text{(xi)}}    &{\sec^{-1} \left(\sec (x) \right)}    &{}\\
{}    &{\text{(vi)}}    &{\cot \left(\cot^{-1} (x) \right)}    &{\text{(xii)}}    &{\cot^{-1} \left(\cot (x) \right)}    &{}\\
\end{array}                   
$

In all the above 12 cases, whatever value we give as input, the output will be that same value.


Now we will see 6 properties of inverse trigonometric functions.

Property I
• This has 3 parts:
\begin{array}{cc}{}    &{\text{(i)}}    &{\sin^{-1}\left(\frac{1}{x} \right)}    {}={}    &{\csc^{-1}x}    &{x \ge 1~\text{or}~x \le -1}\\
{}    &{\text{(ii)}}    &{\cos^{-1}\left(\frac{1}{x} \right)}    {}={}    &{\sec^{-1}x}    &{x \ge 1~\text{or}~x \le -1}\\
{}    &{\text{(iii)}}    &{\tan^{-1}\left(\frac{1}{x} \right)}    {}={}    &{\cot^{-1}x}    &{x > 0}\\
\end{array}

• Let us prove part (i):
$\begin{array}{ll}{}    &{\text{Let}~\sin^{-1} \left(\frac{1}{x} \right)}    & {~=~}    &{y~\color{magenta}{\text{- - - (A)}}}    &{} \\
{\implies}    &{\sin y}    & {~=~}    &{\frac{1}{x}}    &{} \\
{\implies}    &{\frac{1}{\csc y}}    & {~=~}    &{\frac{1}{x}}    &{} \\
{\implies}    &{\csc y}    & {~=~}    &{x}    &{} \\
{\implies}    &{\csc^{-1} x}    & {~=~}    &{y~\color{magenta}{\text{- - - (B)}}}    &{} \\
{\implies}    &{\csc^{-1} x}    & {~=~}    &{\sin^{-1} \left(\frac{1}{x} \right)~\color{magenta}{\text{- - - (C)}}}    &{} \\
\end{array}               
$

◼ Remarks:
• In line (A), we assume that $\sin^{-1}x$ = y
• In line B, we substitute for y, using the assumption in (A).
• When the substitution is done, we get line C. 

• Let us prove part (ii):
$\begin{array}{ll}{}    &{\text{Let}~\cos^{-1} \left(\frac{1}{x} \right)}    & {~=~}    &{y~\color{magenta}{\text{- - - (A)}}}    &{} \\
{\implies}    &{\cos y}    & {~=~}    &{\frac{1}{x}}    &{} \\
{\implies}    &{\frac{1}{\sec y}}    & {~=~}    &{\frac{1}{x}}    &{} \\
{\implies}    &{\sec y}    & {~=~}    &{x}    &{} \\
{\implies}    &{\sec^{-1} x}    & {~=~}    &{y~\color{magenta}{\text{- - - (B)}}}    &{} \\
{\implies}    &{\sec^{-1} x}    & {~=~}    &{\cos^{-1} \left(\frac{1}{x} \right)~\color{magenta}{\text{- - - (C)}}}    &{} \\
\end{array}               
$

◼ Remarks:
• In line (A), we assume that $\cos^{-1}x$ = y
• In line B, we substitute for y, using the assumption in (A).
• When the substitution is done, we get line C.


◼ In the same way, we can prove part (iii) also.
◼ Note:
For each of the three parts in property I, we see acceptable values of x. For example, for part (i), x ≥ 1 or x ≤ -1. We will see the details about such “acceptable values” in higher classes. At present, all we need to know is that, whenever we see $\csc^{-1} (x)$, we can put $\sin^{-1} \left(\frac{1}{x} \right)$ in it’s place.


In the next section, we will see the second and third properties of inverse trigonometric functions.

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