Question

How do you find the value for \[{\csc ^{ - 1}}\left( { - \dfrac{7}{3}} \right)\] ?

Answer 1

Hint: Here we have to determine the value, and the given function is an inverse trigonometry. The csc is a cosecant trigonometry. We simplify the term which is present in parenthesis and by using the table of trigonometry ratios for the standard angles and by the scientific calculator or by the Clark’s table we determine the solution for the question.

Complete step-by-step answer:
The sine, cosine, tangent, cosecant, secant and cotangent are the trigonometry ratios of trigonometry. It is abbreviated as sin, cos, tan, cosec, sec and cot. Here in this question, we have \[{\csc ^{ - 1}}\left( { - \dfrac{7}{3}} \right)\] , where \[{\csc ^{ - 1}}\] represents the inverse of a cosecant function. So we have to find the \[{\csc ^{ - 1}}\left( { - \dfrac{7}{3}} \right)\] .
We consider the table of trigonometry ratios for standard angles.
The table of cosecant function for standard angles is given as

Angle	0	30	45	60	90
csc	\[\infty \]	\[2\]	\[\sqrt 2 \]	\[\dfrac{2}{{\sqrt 3 }}\]	\[1\]

Now consider the given function
 \[{\csc ^{ - 1}}\left( { - \dfrac{7}{3}} \right)\]
As we know the property \[\csc ( - x) = - \csc x\]
When we consider the fraction term and if we simplify the fraction term we obtain 2.3333..
So according to the table the value will be less than 30.
When we find the value by using the scientific calculator or Clark’s table the value of \[{\csc ^{ - 1}}\left( { - \dfrac{7}{3}} \right)\] is \[25.3769\] with 4 decimal places.
So, the correct answer is “ \[25.3769\] ”.

Note: The trigonometry and inverse trigonometry are inverse for each other. The inverse of a function is represented as the arc of the function or the function is raised by the power -1. For the trigonometry and the inverse trigonometry we need to know about the table of trigonometry ratios for the standard angles.