Question

How do you evaluate $ \arcsin (\sin (3\pi )) $ ?

Answer 1

Hint: Here in this question, we have to find the value of a function. The function is a trigonometric function and it also involves inverse trigonometry function. By using the table of trigonometry ratios of standard angles we obtain the required result

Complete step-by-step answer:
The question involves the trigonometry ratio which comes under the concept of trigonometry. The sine function is one of the trigonometry ratios. In the question the function involves the word “arc” the arc is known as inverse. In trigonometry we have a table of trigonometry ratios for the standard angles.
Now consider the question $ \arcsin (\sin (3\pi )) $ , first we find the value of $ \sin (3\pi ) $ . By the table of trigonometry ratios for standard angles. The value of $ \sin (3\pi ) $ is 0. Therefore $ \sin (3\pi ) = 0 $
Now we substitute the value in the above function with $ \arcsin (0) $ . The $ \arcsin (0) $ can be written as $ si{n^{ - 1}}(0) $ .
By the table of trigonometry ratios for standard angles we have $ sin(0) = \pm n\pi $ . In general the value of inverse of a sine function is $ \pm n\pi $ , for n=0,1,2, and so on. Therefore, we get the solution as $ \pm n\pi $ .
Hence $ \arcsin (\sin (3\pi )) = \pm n\pi $ .
We can also solve the above function by another method.
The function $ \arcsin (\sin (3\pi )) $ contains trigonometry function and inverse trigonometry function of trigonometry ratio for sine function. The sine function and the inverse sine function will cancel each other.
 Hence by cancelling the trigonometry ratios we obtain the final result as $ 3\pi $ . The $ 3\pi $ contains in the $ n\pi $ .
Hence, we have evaluated the given function and obtain the result.
Therefore $ \arcsin (\sin (3\pi )) = \pm n\pi $ or $ \arcsin (\sin (3\pi )) = 3\pi $
So, the correct answer is “ $ 3\pi $ ”.

Note: In trigonometry and inverse trigonometry we have a table for trigonometry ratios for standard angles. By using the table, we can determine the values. The inverse for the trigonometry ratio is represented by arc or trigonometry ratio is raised by -1. Hence we can solve these types of questions by knowing the table of trigonometry ratios for standard angles.