Hey, everyone. In this problem, we're asked to evaluate the expression the inverse sine of the sine of pi over 6. Now, here, you might be tempted to just cancel the inverse sine with the sine and end up with your answer of pi over 6. But remember, we have to be extra careful when working with these composite trig functions because our inverse trig functions are only defined for a particular interval. So let's go ahead and break this down the way we would any composite trig function starting with that inside function, the sine of pi over 6.
Now coming over here to my unit circle, the sine of pi over 6 is equal to 1/2. So that inside function gives me a value of 1/2, and now I'm just left to find the inverse sine of that 1/2. Now remember, when working with inverse trig functions, we can also think of this as, okay, the sine of what angle is equal to 1/2? But that angle can only be within our specified interval for the inverse sine, which happens to be from a negative pi over 2 to a positive pi over 2. So you only want an angle within this interval for which the sine is equal to 1/2.
Now inside of this interval, where is my sine equal to 1/2? Well, the sine of pi over 6 is equal to 1/2, and that is within my interval. So that actually gives me a final answer of pi over 6. Now here you might be confused because I just told you that you can't cancel the inverse sine with the sine, but here we actually could. Now the reason that we could do that is because our angle from the beginning was within our specified interval.
So when that does happen, you actually are able to effectively cancel the inverse sine with the sine. But you always want to be extra careful when working with these because, remember, that isn't always true. So always, always double-check your interval and make sure you're getting a value only within your specified interval. Thanks for watching, and let me know if you have questions.