In the previous section, we saw cos-1 function. In this section, we will see cosec-1 function.
Some basics can be written in 8 steps:
1. Consider the cosec function:
f(x) = cosec x
◼ For this function, we must choose the input values carefully. It can be written in 3 steps:
(i) We know that: $\csc x = \frac{1}{\sin x}$.
• The denominator should not become zero. That means, sin x should not become zero.
(ii) We know that:
♦ sin 0 = 0
♦ sin π = 0
♦ sin (-π) = 0
♦ sin 2π = 0
♦ sin (-2π) = 0
♦ sin 3π = 0
♦ so on . . .
• So we can write:
The input x should not be equal to nπ, where n is an integer.
(iii) Thus we get the domain for f(x) = cosec x as:
set {x : x∈R and x ≠ nπ, n∈Z}
• That means, x can be any real number except nπ, where n is an integer.
◼ Similarly, we must have a good knowledge about the codomain. It can be written in 3 steps:
(i) We know that, output of sin x will lie in the interval [-1,1].
• That means, output of sin x will be any one of the four items below:
♦ -1
♦ a -ve proper fraction
♦ a -ve proper fraction
♦ +1
(Though zero lies in the interval [-1,1], we are not allowing sin x to become zero. We achieve this by avoiding nπ as input x values)
(ii) Based on the above "possible outputs of sin x", we can write the "possible outputs of cosec x":
♦ If it is -1, the output will be -1.
♦ If it is a -ve proper fraction, the output will be real number smaller than -1.
♦ If it is a +ve proper fraction, the output will be real number larger than 1.
♦ If it is +1, the output will be +1.
• We see that, cosec x will not give outputs in the interval (-1,1). Note that, for writing this interval, we use '()' instead of '[]'. This is because, -1 and +1 are not included in the interval. Those two values can become outputs.
(iii) Thus we get the codomain:
R − (−1,1)
• We saw the above details in class 11. We saw a neat pictorial representation of the above details in the graph of the cosec function. It is shown again in fig.18.10 below:
 |
| Fig.18.10 |
• Let input x = $\frac{\pi}{2}$.
♦ Then the output will be $f \left(\frac{\pi}{2} \right)~=~\csc \left(\frac{\pi}{2} \right)~=~1 $
• Let input x = $\frac{5 \pi}{2}$.
♦ Then the output will be $f \left(\frac{5 \pi}{2} \right)~=~\csc \left(\frac{5 \pi}{2} \right)~=~1 $
• Let input x = $\frac{-3 \pi}{2}$.
♦ Then the output will be $f \left(\frac{-3 \pi}{2} \right)~=~\csc \left(\frac{-3 \pi}{2} \right)~=~1 $
(We can cross check with the graph and confirm that the above inputs and outputs are correct)
•
We see that, more than one input values from the domain can give the
same output. So the cosec function is not a one-one function.
3. Suppose that, we restrict the input values.
• That is, we take input values only from the set $\left[\frac{- \pi}{2}, \frac{\pi}{2} \right]$.
♦ But zero lies in this interval. Zero cannot be an input.
♦ So the modified set is: $\left[\frac{- \pi}{2}, \frac{\pi}{2} \right]~-~\{ 0 \}$
• We already saw the codomain. It is: R − (−1,1)
• Then we will get the green curves shown in fig.18.11 below:
 |
| Fig.18.11 |
4. Next, we have to prove that, the green portion is onto.
• For that, we can consider any y value from the codomain R − (−1,1).
• There will be always a x value in $\left[\frac{- \pi}{2}, \frac{\pi}{2} \right]~-~\{ 0 \}$, which will satisfy the equation y = cosec x.
• So the green portion is onto.
5. We see that, the green portion is both one-one and onto.
• We can represent this function in the mathematical way:
$\text{cosec}:~ \left[\frac{- \pi}{2}, \frac{\pi}{2} \right]~-~\{ 0 \}~\to~R~-~(-1,1)$, defined as f(x) = cosec x.
6. If a function is both one-one and onto, the codomain is same as range.
• So we can write:
For this function,
♦ the domain is $\left[\frac{- \pi}{2}, \frac{\pi}{2} \right] - \{ 0 \}$
♦ the range is R − (−1,1)
7.
We know that, if a function is one-one and onto, it will be invertible.
We have seen the properties of inverse functions. Let us apply those
properties to our present case. It can be written in 4 steps:
(i) If y = f(x) = cosec x is invertible, then there exists a function g such that: g(y) = x
(ii) The function g will also be one-one and onto.
(iii) The domain of f will be the range of g. So the range of g is $\left[\frac{- \pi}{2}, \frac{\pi}{2} \right]~-~\{ 0 \}$.
(iv) The range of f will be the domain of g. So the domain of g is R − (−1,1)
(iv) The inverse of cosec function is denoted as cosec-1. So we can define the inverse function as:
$\csc^{-1}:~ R − (−1,1)~\to~\left[\frac{- \pi}{2}, \frac{\pi}{2} \right]~-~\{0 \}$, defined as x = g(y) = cosec-1 y.
8. Let us see an example:
• Suppose that, for the inverse function, the input y is 2
• Then we get an equation: $x ~=~\csc^{-1} \left(2 \right)$
• Our aim is to find x. It can be done in 4 steps:
(i) $x ~=~\csc^{-1} \left(2 \right)$ is an equation of the form $x ~=~f(y)~=~\csc^{-1} \left(y \right)$
(ii) This is an inverse trigonometric function, where input y = 2.
•
Based on the inverse trigonometric function, we can write the original trigonometric function:
$y ~=~ f(x) ~=~\csc x$
•
In our present case, it is:
$y~=~2~=~\csc x$
(iii) So we have a trigonometric equation:
$\csc x~=~2$
•
When we solve this equation, we get x.
(iv) We have seen the method for solving trigonometric equations in class 11.
•
In the present case, we do not need to write many steps. We already
know that, $\csc \left(\frac{\pi}{6} \right)~=~2$
•
So we can write:
$x ~=~\frac{\pi}{6}$
The
above 8 steps help us to understand the basics about cosec-1 function. Now we will see a few more details. It can be
written in 4 steps:
1. We saw that, the cosec function is not a one-one
function. But to make it one-one, we restricted the domain to $\left[\frac{- \pi}{2}, \frac{\pi}{2} \right] - \{ 0 \}$.
2. There are other possible “restricted domains” available.
• $\left[\frac{- 3 \pi}{2}, \frac{- \pi}{2} \right] - \{- \pi \}$ is shown in magenta color in fig.18.12 below:
 |
| Fig.18.12 |
• $\left[\frac{\pi}{2}, \frac{3 \pi}{2} \right] - \{\pi \}$ is shown in cyan color in fig,18.12 above.
3. There are infinite number of such restricted domains possible.
• We say that:
♦ Each restricted domain gives a corresponding branch of the cosec-1 function.
♦ The restricted domain $\left[\frac{- \pi}{2}, \frac{\pi}{2} \right] - \{ 0 \}$ gives the principal branch of the cosec-1 function.
4. In class 11, we plotted the cosine function. Now we will plot the inverse. It can be done in 4 steps:
(i) Write the set for the original function f. It must contain a convenient number of ordered pairs.
• $\left(\frac{\pi}{6} , 2 \right)$ is an example of the ordered pairs in f.
(To get a smooth curve, we must write a large number of ordered pairs)
(ii) Based on set f, we can write set g.
This is done by picking each ordered pair from f and interchanging the positions.
•
For example, the point $\left(\frac{\pi}{6} , 2 \right)$ in
the set f will become $\left(2 , \frac{\pi}{6} \right)$ in
set g.
• Thus we will get the required number of ordered pairs in g.
(iii) Mark each ordered pair of g on the graph paper.
• The first coordinate should be marked along the x-axis.
• The second coordinate should be marked along the y-axis.
(iv) Once all the ordered pairs are marked, draw a smooth curve connecting all the marks.
• The smooth curve is the required graph. It is shown in fig.18.13 below:
 |
| Fig.18.13 |
(v) Unlike the graphs of sin-1 and cos-1, the graph of cosec-1 is not smaller in width. This is because:
Any real number except those in the interval (-1,1), can be used as input values. That means, the graph can extend upto -∞ towards the left and upto +∞ towards the right.
• The graph is larger in height because:
♦ Depending upon the branch, values upto +∞ or –∞ can be obtained as output values.
• The green curve is related to the $\left[\frac{- \pi}{2}, \frac{\pi}{2} \right] - \{ 0 \}$ branch.
• The cyan curve is related to the $\left[\frac{\pi}{2}, \frac{3 \pi}{2} \right] - \{\pi \}$ branch.
• The magenta curve is related to the $\left[\frac{- 3 \pi}{2}, \frac{- \pi}{2} \right] - \{- \pi \}$ branch.
In the next section, we will see sec-1 function.
Previous
Contents
Copyright©2024 Higher secondary mathematics.blogspot.com
No comments:
Post a Comment