 We've reformulated the trigonometric functions in terms of angles in a unit circle. Let's see what else we can do with this change of viewpoint. And this relates to the following geometric transformations. Given a point x, y, a reflection across the x-axis will take it to the point with coordinates x, negative y. Likewise, if we reflect it across the y-axis, we go to the point minus x, y. And a reflection across the origin will take us to the point with coordinates minus x, minus y. For example, suppose that the cosine of theta is three-fifths, and theta is an angle of the first quadrant. Can we find cosine of minus theta, cosine of 180 degrees minus theta, and tangent of 180 degrees plus theta? And the answer is? Well, I hope we can find it, because that's what the question is asking us to do. To begin with, remember that if theta is an angle in standard position, the terminal side of the angle passes through x, y on the unit circle, x squared plus y squared equals 1, where x is the cosine, and y is the sine. So since theta is in the first quadrant, we can draw it. Since the cosine is three-fifths, the terminal side of this angle passes through a point b on the unit circle with coordinates x, y, where x is three-fifths. And y can be found by solving x squared plus y squared equals 1. So p has coordinates three-fifths, y, where we can find y if we need to. Since minus theta indicates a rotation in the opposite direction, we can also draw an angle minus theta. The terminal side of this angle passes through a point b prime that is a reflection across the x-axis of the point b. So that tells us the coordinates of this point will be three-fifths, negative y. But our x value is the cosine. So since b prime represents an angle of minus theta, we can read this off, cosine of minus theta is three-fifths. Now this is an example of a very useful theorem. For any angle theta, cosine theta is cosine of minus theta. And here's an important point to remember. Don't memorize theorems. Understand concepts instead. In fact, you're better off letting this theorem fade from your memory. Instead, understand the process that gave us this result, that theta and minus theta indicate rotations in the opposite direction, that we have a reflection of our point across the x-axis, and that cosine is the x-coordinate. So for example, if we want to find the cosine of 180 degrees minus theta, we have to think about this angle 180 degrees minus theta. And the angle 180 minus theta can be found by a counterclockwise rotation of 180 degrees, that's halfway around, followed by a clockwise rotation back of theta, that's back theta. So if we look at this, the terminal side of the angle 180 degrees minus theta will pass through some point b' that is the reflection of b across the y-axis. So the coordinates of p' will be minus three-fifths y. And since the x-coordinate is the cosine, we can read this off, cosine of 180 minus theta is minus three-fifths. And this is an example of a theorem. But again, the theorem isn't worth memorizing. What's more important is understanding this concept of measuring the angle by a rotation, then locating the terminal side by a reflection, and remembering that the cosine is the x-coordinate. How about the tangent of 180 degrees plus theta? Well again, the angle of 180 plus theta can be found by rotating counterclockwise a half turn, and then rotating an additional amount theta. And we see that the terminal side of the angle 180 plus theta passes through a point p' that is the reflection of the point p across the origin. And so the coordinates of this point will be minus three-fifths minus y. Since tangent of theta is y over x, we need to find y. Since this is a point on the unit circle, we know that x squared plus y squared equals 1, and we know the value of x, so let's go ahead and solve for the value of y. Since x, y is in the first quadrant, we know that y has to have the positive value, and that means the reflection across the origin is going to have coordinates minus three-fifths minus four-fifths. And tangent is our y-coordinate divided by our x-coordinate. So tangent is going to be minus four-fifths over minus three-fifths, which will be four-thirds. And this is an example of a theorem, but who cares because you're not going to memorize the theorem? You're going to understand the concept of a rotation and a reflection. Of course, suppose we know that sine of theta is five-thirteenth, and theta is the first quadrant angle. Let's find cosine of 90 degrees minus theta. So a useful thing to remember is that success in mathematics and life is all about habits. By now, you should have developed a habit that any time you have a trigonometric problem, you should... Since theta is the first quadrant angle, we know that its terminal side passes through some point B in the first quadrant, and the sine gives us the y-coordinate. Putting this all together, since x, y is on the unit circle, and we know sine is five-thirteenths, then y is five-thirteenths, and x squared plus y squared equals one. Solving this equation, and as an equation, x has two possible values, plus or minus twelve-thirteenths, but because our point is in the first quadrant, we know that x has to be positive. So this tells us the terminal side of theta passes through the point, twelve-thirteenths, five-thirteenths, and if it's not written down, it didn't happen. Now the angle 90 degrees minus theta can be found by rotating a quarter-turn counterclockwise that's 90 degrees, and then rotating clockwise by theta, and the terminal side of 90 degrees minus theta will pass through the point B' on the unit circle. Again, another useful habit to get into when dealing with trigonometry, and really all mathematics after this point, is to look for the right triangle. One of the right triangles is this one, and that's the one that gave us sine theta equals five-thirteenths. This other angle actually gives us two right triangles. There's this one, which is the obvious one, but there's a second right triangle that'll be the important one, and it's here. Now because both of these angles are theta and both of these angles are right angles, and this length is one, these two right triangles are congruent, and that means these two lengths are equal, and these two lengths are equal. Well, this length is our x-coordinate of the point P, so that's twelve-thirteenths, which means that this length has to be twelve-thirteenths. But that's the y-coordinate of the point B' so we know that the y-coordinate must also be twelve-thirteenths. Likewise, this length is our y-coordinate, so it must be five-thirteenths, which means that this length is also five-thirteenths. But that's the x-coordinate of our point B' and so we see that the terminal side of the angle 90-theta passes through the point five-thirteenths, twelve-thirteenths. But the cosine is the x-coordinate, so the cosine of 90-theta must be five-thirteenths. And again, this is an example of a theorem that it's not worth memorizing because it's more important to understand the concept that produced this theorem.