 So, given an angle, we can find, using calculator, table, random guesses, or drawing a picture, the values of sine, cosine, and tangent, as well as the values of secant, cosecant, and cotangent. But sometimes we want to go backwards. Ultimately, this will come back to solving equations, because remember, equation solving requires us to apply inverse functions. So if I want to solve x plus seven equals fifteen, the operation is addition, so we apply the inverse operation, subtraction. Or if I want to solve three x equals eleven, the operation is multiplication, so we apply the inverse operation, division. If I have x squared equals thirty-seven, the operation is squaring, so we apply the inverse operation, the square root. If I want to solve e to the x equals five, the operation is exponentiation, so we apply the inverse operations, and we hit both sides with the log. And so now we have these new functions, these trigonometric functions, and we could have equations with them. Suppose you have the equation sine of theta is equal to y. To solve it, you'd need the inverse sine operation. Well let's get into the mind of a mathematician. Don't worry, it's not too scary a place. And let's define the inverse sine function, which we'll write as arc sine z to give the angle theta where the sine of theta is equal to z. Actually, we'll need to modify this definition several times before we're done, so we'll give it a version number. We'll call this version 1.0. So let's see what we can do with this. Well let's try to find the arc sine of one. Definitions are the whole of mathematics. All else is commentary, so let's pull in our definition version 1.0, and the arc sine of z is going to give us the angle theta where the sine of theta is equal to z. So that means by our definition, we want to find theta where the sine of theta is equal to one. Now from our unit circle definition, we know that the sine of theta is equal to the y coordinate. So we want to find the points on the unit circle where y is equal to one. So let's do this the hard way. Wait, wrong script. Let's do this the easy way and draw a picture. We want to draw our unit circle and find the points where y is equal to one. But the only point on the unit circle where y is equal to one is at the top, 0, 1. And remember the arc sine of z gives us an angle theta, so we have to figure out the angle that this top point corresponds to. And if we think about it, we see that we need to rotate a quarter-turn counterclockwise to get there, and so this corresponds to an angle of 90 degrees. So a politician or a demagogue would be happy and say, Ha, we've solved the problem. Everything can be solved by this very first thing that popped into my head. And what tends to make mathematicians unpopular at cocktail parties is they ask things like, well, what happens if? So what can we say about theta where sine of theta is equal to one-half? Well, again, on the unit circle, sine of theta is our y value, and equals means interchangeable if sine of theta equals one-half, y is equal to one-half. So we want to find the point of the unit circle where y is equal to one-half. So let's draw a picture and find this point. Except we see there are two points where this occurs. So there are two values theta for which sine of theta is equal to one-half. There's a first quadrant angle theta. But if we keep going, we'll get to a second quadrant angle theta. Which one do we want to use? So we see there are two angles theta where sine of theta is equal to one-half. So which one do we choose? The theta in the first quadrant or the theta in the second quadrant? We choose the first quadrant angle. So this means we need to modify our definition slightly. So we're still going to define the inverse sine function to give the angle theta where the sine of theta is equal to z. But if there's more than one angle theta that satisfies this, we choose the angle in the first quadrant. Well, what about sine of theta equals minus one-half? So again, on the unit circle, sine of theta equals y equals means replaceable. So in place of sine of theta, I can write minus one-half. And so y equals minus one-half. So we want to find the point on the unit circle where y equals negative one-half. So we'll draw a unit circle and try to find these points. We see there are two points where this occurs. One is a rotation into the third quadrant. And another is a rotation into the fourth quadrant. So there are two values theta for which sine of theta is minus one-half. So we'll pick the angle in the first quadrant. Uh, wait a minute, neither of these is in the first quadrant. And that means we'll have to modify our definition once again. And this is just a little change. If there's more than one angle, we choose the angle in the first or fourth quadrant. So far, so good. But there's one additional complication. Even though our angle may be in the first quadrant, we can measure it in several different ways. First, we might have a counterclockwise rotation of theta. Maybe we could go a full turn counterclockwise than a counterclockwise rotation of theta. Or maybe we could do a full turn clockwise than an additional rotation counterclockwise of theta. And in fact, we could rotate any number of times counterclockwise or clockwise before performing a rotation of theta. But we don't want to. And so this gives us a new definition. One more revision. Again, we're still going to choose that angle in the first or fourth quadrant, but we'll use the least rotation possible. In other words, we're not going to rotate around counterclockwise or clockwise until we get there, but we'll just rotate once. Well, there's a little bit more fine print we have to read. If our angle is in the fourth quadrant, there are two ways we can measure it. First, we've been measuring our angles positively through a counterclockwise rotation. But the other possibility is we could measure it negatively through a clockwise rotation. So we have to choose, and we'll choose to use the negative rotation if we have to. And so our final definition, arc sine of z, is going to give us an angle in the first or fourth quadrant using the least rotation possible from the positive x-axis. Similarly, we have the inverse cosine function, written arc cos z, which gives us the angle theta where cosine of theta is equal to z. And this time, if there's more than one angle that satisfies this, we choose the angle in the first or second quadrant and use the least possible rotation from the positive x-axis. The inverse tangent function, written arc tan, is the same angle theta who are tangent of theta is equal to z. And this time, we'll use the first or fourth quadrant with the least possible rotation. And there's also arc secant, arc cotangent, and arc cosecant of z, but no one really cares about them. What is important to remember is that the angles returned by the inverse trigonometric functions will be in a quadrant to the right or above the origin. Now, one caution on notation. We also write arc sine z is sine negative 1z, arc cosine, cosine negative 1z, and arc tan, tan negative 1z. But it's important to remember the minus one is not an exponent. It refers to the inverse function. So, for example, in what quadrant is the inverse cosine of minus one-fifth? So remember, cosine minus one, negative one-fifth, is the same as arc cosine of minus one-fifth. So, according to our definition, the arc cosine of minus one-fifth is the angle in the first or second quadrant whose cosine is minus one-fifth. Since cosine theta is the x-coordinate, we need a point on the unit circle with a negative x-coordinate. Now, there are two points on the unit circle that have an x-coordinate of negative one-fifth. They're here and here. But the arc cosine itself has to be a first or second quadrant angle, and so that means arc cosine of minus one-fifth will be an angle in the second quadrant.