 The reason that we've developed the inverse trigonometric functions is to be able to solve trigonometric equations. But one important distinction about trigonometric equations is that in general they have an infinite number of solutions. So how do we find all of them? For example, let's see if we can find all solutions to cosine theta equals four fifths. Now let's take this equation apart. We want to find an angle whose cosine is four fifths. Since you want to make this problem as hard as possible, you don't draw a picture. Wait a minute, wrong script. Since you want to make this problem as easy as possible, you draw a picture. So remember that on the unit circle, the cosine corresponds to the x-coordinate of a point. And we see that if we draw a picture, there are two points where x is equal to four fifths, and this gives us two angles whose cosine is four fifths. We can rotate by a positive amount to get to the first point, or rotate by a negative amount to get to the other point. Now we can find one of these angles using the inverse cosine function. A calculator, computer algebra system, spreadsheet, or random passerby will tell you that the inverse cosine of four fifths is about 37 degrees. And so this gives us one solution. But remember, the inverse cosine function will give us an angle in the first or second quadrant. And so 37 degrees will be the angle in the first or second quadrant whose cosine is four fifths. What about this angle in the fourth quadrant? How can we find that? The thing that recognizes the other angle is a reflection of this angle across the x-axis. So its measure will be minus 37 degrees. Now, while there's nothing wrong with an angle of minus 37 degrees, it's preferable to express this without a negative angle. So one way of getting that is if we go once around from minus 37 degrees, we end at 323 degrees. And so this gives us a second solution. But wait, there's still more. We can add or subtract any integer multiple of 360 degrees to get additional solutions. And that's because after we've rotated 37 degrees, if we make a full rotation after that, our cosine is still the same. Likewise, if we get to 323 degrees, if we make full rotations after that, we still have the same cosine. And so all solutions are going to be expressed as 37 degrees plus or minus any multiple of 360 degrees, and 323 degrees plus or minus any multiple of 360 degrees. Or suppose we want to find all solutions to tangent theta equals five-thirds. We could make this problem difficult by not drawing a picture, but why would we want to? So remember, on the unit circle, the tangent of an angle corresponds to the value of y over x. Now the picture is primarily a way to organize our information, so it doesn't have to be perfect. But it's nice if it's at least somewhat close. So let's think about our given information a little bit. Now since tangent of theta is equal to y over x, then if tangent of theta is five-thirds, it must be that y is greater than x. So this point B on the unit circle must be somewhere above the line y equals x. So maybe we'll drop it right around here. So one solution will be theta equals arc tangent of five-thirds. So a computer, calculator, or random passerby will tell us that arc tangent of five-thirds is about 1.0304. Uh, wait a minute, since we drew a picture, we know that whatever this is, this can't be an angle in degrees. One degree is a very tiny angle. And it's important to remember, if you use power tools without taking precautions, you are guaranteed to have an accident. In this case, the important thing to remember is that since measuring angles in radians is about a billion, billion, billion, billion times more useful than measuring angles in degrees, most computers, calculators, computer algebra systems, and spreadsheets, and random passersby will give you the inverse trigonometric function values in radians unless you ask otherwise. And so we need to make sure that our calculator, computer, CAS, or random stranger is set to degree mode. And if we do that, they'll tell us that the inverse tangent of five-thirds is about 59 degrees. Well, that gives us one solution. Now, since tangent is y over x, changing the sign of both x and y will leave the tangent unchanged. And so we can reflect this across the origin, which corresponds to a rotation of an additional 180 degrees. So another solution is going to be 59 degrees plus 180 degrees. And to either of these, we can go around any additional number of turns. So to either of these solutions, we can add any multiple of 360 degrees. And more simply, we can do any number of half turns from 59 degrees. So we can also express our solutions as 59 degrees plus any number of 180 degree turns.