Do I have to worry about the calculation shown in this picture? I mean finding out an angle from its sine or cosine that I calculate? I am worried bacause it might show in the exam and calculators needed to solve it are not allowed

Knowing this formula is relevant for the IMAT, but you will never get angle entities which are impossible to calculate their exact trigonometric values without a calculator.

Generally, IMAT calculation questions are always based on rounded or known applicable values.