sine of the inverse sine of 1.325

This is a domain question in disguise. Most high school and college math textbooks introducing inverse trigonometric functions would indicate that the value of sin(sin^{−1}(1.325)) does not exist because sin^{−1}(1.325) doesn't exist.

The sine of any real angle must fall between −1 and 1. This means that the inverse sine function is only permitted to have values between −1 and 1 as its input. Using math vocabulary, the domain of the inverse sine function within the real numbers is [−1,1]. Trying to find sin^{−1}(1.325) within the real numbers is like trying to find √−1 within the real numbers. It doesn't exist because you need a complex number to express the answer.

Of course, if your class is at a more advanced level and is covering the trigonometry of complex numbers then it actually is true that sin(sin^{−1}(1.325)) = 1.325 within complex numbers, but then the question wouldn't be a domain question in disguise.

## Comments

Thanks Carlos!