 We can expand our ability to integrate and differentiate if we look at the inverse trigonometric functions. The inverse trigonometric functions emerge as follows. Suppose y is some trigonometric function of x. The inverse trigonometric function y equals inverse trigonometric function of x satisfies the relationship x equals the trigonometric function of y. But unless we're careful, this is not going to be a function. And the reason is that the trigonometric functions are periodic. So for example, suppose I want to find three solutions to the inverse sine of zero equals y. By definition, if the inverse sine of zero is equal to y, then zero is equal to the sine of y. And so let's try and find solutions. If y is equal to zero, then sine of zero is zero, so y equals zero is a solution. We can find another solution. If y equals two pi, then sine of two pi is zero. And if y equals negative pi, sine of negative pi is also equal to zero. So y equals zero, two pi, and negative pi are three different solutions. And in fact, there are many more. And so because the inverse sine of x equals y will have many values of y for any given x, the inverse sine of x will not be a function of x unless we identify which of these many values it returns. We sometimes say we must choose a branch of the inverse relation. This isn't actually a new idea. You've already seen this before in the context of square roots. If y squared equals x, we say that y is a square root of x. So five and negative five are both square roots of 25 since five squared is 25 and negative five squared is 25. But in order for square root of x, this algebraic symbol to be a function, we need to choose a branch. Which of these square roots is this thing, square root of x, going to give us? And so for y equals square root of x, we choose the positive number that satisfies y squared equals x. And so if I write square root of 25, this can only be equal to five. So let's consider those branches for arc sine of x. Suppose sine of y equals x. We want to find y equals arc sine x as a function. So we might begin as follows. We know that x has to be between negative one and one. So we want to limit y to an interval that, first of all, includes a solution to sine y equals x for x between negative one and one. And also that if a solution exists, only one solution is in that interval. So let's consider a few intervals and see which one will work for us. So for example, how about zero less than or equal to y less than or equal to pi? Will this work as an interval? And unfortunately, this won't. This won't allow us to solve sine y equals negative one. There's no solution in that interval to this equation. Well, how about y between zero and two pi? While this meets that first requirement that we include all solution to sine y equals x for x between negative one and one, it misses the second requirement that only one solution is in the interval. There are multiple solutions to things like sine y equals zero. So how about y between minus pi and pi? And we meet that first requirement that sine y equals x will have a solution. But again, we fail that second requirement that there are multiple solutions to equations like sine of y equals zero. Well, third time's the charm, so if we don't find it by the third time, we should give up and join the circus. Okay, let's try one more time. How about y between minus pi over two and pi over two? We see that this will include a solution to sine y equals x for any x between negative one and one. Moreover, if a solution exists, only one solution can be found in this interval. And so this suggests the following definition for arc sine. We'll say that y is the arc sine of x, where y is between minus pi over two and pi over two, and sine of y is equal to x. By a similar argument, if we want cosine of y to be x, we want to define y equals arc cosine of x as a function. And in this case, we'll choose the interval y is between zero and pi. And likewise, if tan y equals x, then y equals the arc tangent of x, where y once again falls between minus pi over two and pi over two. One more note. The more important something is, the more names we typically have for it, because different groups of people find it important and they come up with their own names for it. So even though we use arc cosine arc sine arc tangent, an alternate set of notation is sine inverse for arc sine, cosine inverse for arc cosine, and tangent inverse for arc tangent. The notation for the inverse trigonometric functions is not consistent. So we're in calculus, and the first question that we always ask whenever we get presented with a shiny new function is, what's the derivative? So we might start with a definition. If y equals arc sine of x, then sine of y is equal to x. And so we can use implicit differentiation. Now, while I do get the answer, derivative equals one of our cosine y, since the function is in terms of x, its derivative should also be in terms of x. We don't want to leave our derivative in terms of y. So that means we need to do some algebra and or trigonometry. And one of the things we know is the Pythagorean identity, cosine squared y plus sine squared y equals one. We can solve this equation for cosine y. Now remember, the square root gives us the positive number whose square is cosine y, but I have the possibility that there may be a negative number in there as well. So I do have to indicate plus or minus square root. And because I know that sine of y is equal to x, this radicand becomes one minus x squared. However, there's a problem. We can't use plus or minus in a function, as this would give us more than one output. We have to make a choice. So again, we'll go back to our definition of what arc sine is. And remember that if y is equal to arc sine of x, then we know that y is between minus pi over two and pi over two. And this means that the cosine of y is going to be positive. And so that means while cosine of y could be plus or minus one over x squared, what it actually is is the positive square root, which we can then substitute back into our derivative and get our form of the derivative of arc sine of x.