Question

How do you plug the inverse $\csc \left( -7/3 \right)$ into the calculator$?$

Answer 1

The concept used to solve the problem will be to reverse the trigonometric function. The trigonometric function $\csc$ is also written as $\text{cosecant}$. We need to check the domain of the inverse of the trigonometric function $\csc$. The domain of inverse $\csc$ which is represented by ${{\csc }^{-1}}$, is $\left( -\infty ,\left. -1 \right] \right.\cup \left[ 1,\left. \infty \right) \right.$.

Complete step by step solution:
The question asks us to find the inverse of $\text{cosec}\left( -7/3 \right)$ . To solve this question we can change the trigonometric function $\csc$ into any other trigonometric function. We need to firstly see the domain of the inverse function $\csc$. The domain of the trigonometric inverse of $\csc$ should be greater than equal to $1$ and less than equal to $-1$ . Mathematically for a function ${{\csc }^{-1}}a$ the value of its domain is $a$ which will be $a\ge 1$ and $a\le 1$ . Since, in this question $a=\dfrac{-7}{3}$ , this means the angle will lie between $-\dfrac{\pi }{2}$ to $\dfrac{\pi }{2}$ . On analysing the trigonometric function $\csc$ we find that it relates the best with the trigonometric function $\sin$. Now the relation between the function is:
${{\csc }^{-1}}a={{\sin }^{-1}}\left( \dfrac{1}{a} \right)$
On putting the values given in the question we get:
$\Rightarrow {{\csc }^{-1}}\left( \dfrac{-7}{3} \right)={{\sin }^{-1}}\left( \dfrac{1}{\dfrac{-7}{3}} \right)$
The value in the domain of ${{\sin }^{-1}}$ will reciprocal itself as $1$ divided by the fraction results in interchanging the numerator and denominator.
$\Rightarrow {{\csc }^{-1}}\left( \dfrac{-7}{3} \right)={{\sin }^{-1}}\left( \dfrac{-3}{7} \right)$
We know that the domain of ${{\sin }^{-1}}$ trigonometric functions from $-1$ to $+1$ including $-1$ and $1$, which is represented as $\left[ -1,1 \right]$. This means ${{\sin }^{-1}}\left( \dfrac{-3}{7} \right)$ is valid.
$\therefore$ We can find the value of ${{\csc }^{-1}}\left( \dfrac{-7}{3} \right)$ by changing it to ${{\sin }^{-1}}\left( \dfrac{-3}{7} \right)$ using calculator.

Note: We use the similar idea to find the inverse of trigonometric function $\text{secant}$ and $\text{cotanget}$ on many calculators. We should know the range of the trigonometric function and inverse trigonometric function as it helps us in knowing whether the function is used correctly or not. Sometimes to find the value of a certain trigonometric function it is changed into the different trigonometric is applied which makes solving easier.