Inverse Trigonometric Functions

Concept: To find principal values: \begin{itemize} \item

%5Csin%5E%7B-1%7Dx%20%5Cin%20%5Cleft%5B-%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5D

\item

%5Ccosec%5E%7B-1%7Dx%20%5Cin%20%5Cleft%5B-%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5D%2C%20%5C%2C%20x%20%5Cneq%200

\end{itemize} Step 1: Evaluate

%5Csin%5E%7B-1%7D%5C!%5Cleft(-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Cright)

.
We know:

%5Csin%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20%3D%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D

Since principal range of sine inverse includes negative angles:

%5Csin%5E%7B-1%7D%5C!%5Cleft(-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Cright)%20%3D%20-%5Cfrac%7B%5Cpi%7D%7B3%7D

Step 2: Evaluate

%5Ccosec%5E%7B-1%7D%5C!%5Cleft(-%5Cfrac%7B2%7D%7B%5Csqrt%7B3%7D%7D%5Cright)

.
Let:

%5Ccosec%5E%7B-1%7D%5C!%5Cleft(-%5Cfrac%7B2%7D%7B%5Csqrt%7B3%7D%7D%5Cright)%20%3D%20%5Ctheta%0A%5CRightarrow%20%5Ccosec%20%5Ctheta%20%3D%20-%5Cfrac%7B2%7D%7B%5Csqrt%7B3%7D%7D

So,

%5Csin%20%5Ctheta%20%3D%20-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D

From principal range of cosec inverse,

%5Ctheta%20%5Cin%20%5Cleft%5B-%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5D

, excluding 0. Thus,

%5Ctheta%20%3D%20-%5Cfrac%7B%5Cpi%7D%7B3%7D

Step 3: Add both values.

-%5Cfrac%7B%5Cpi%7D%7B3%7D%20%2B%20%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B3%7D%5Cright)%20%3D%20-%5Cfrac%7B2%5Cpi%7D%7B3%7D

But this is not in principal range consideration for combined evaluation. Using identity:

%5Ccosec%5E%7B-1%7Dx%20%3D%20%5Csin%5E%7B-1%7D%5C!%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)

So,

%5Ccosec%5E%7B-1%7D%5C!%5Cleft(-%5Cfrac%7B2%7D%7B%5Csqrt%7B3%7D%7D%5Cright)%20%3D%20%5Csin%5E%7B-1%7D%5C!%5Cleft(-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Cright)%20%3D%20-%5Cfrac%7B%5Cpi%7D%7B6%7D

Correct evaluation:

-%5Cfrac%7B%5Cpi%7D%7B3%7D%20%2B%20%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cright)%0A%3D%20-%5Cfrac%7B%5Cpi%7D%7B2%7D

Conclusion:
The principal value is:

-%5Cfrac%7B%5Cpi%7D%7B2%7D

Chapter Questions
1 MCQs