 The trigonometric functions sine, cosine, and tangent are all defined in terms of lengths of the sides of a right triangle, but sometimes that's not enough. In life a lot of interesting stuff happens when we ask nonsense questions. Sometimes it's good. What happens if I put peanut butter on a chocolate bar? Other times, not so much. What happens if I put fish fins on a car? The important thing is the only way to figure out whether it's good or bad is to try it and find out. For example, what is the sine, cosine, and tangent of 120 degrees? So let's pull in our definition. So here we'd want to start with a right triangle where one angle has a measure of 120 degrees and there's a problem here no such triangle can exist. And so we might say that the sine, cosine, and tangent are undefined. But what fun would that be? For convenience we can move any angle into standard position. And that's going to require the following. First the vertex should be at the center, the origin. So we'll throw down our coordinate axes and move our vertex to the origin. One leg should be on the positive x-axis, we'll call this the initial side. So we'll rotate the angle until one leg is on the positive x-axis. And remember the measure of the angle is the amount of rotation necessary to go from the initial side to the other side of the angle, which we call the terminal side. Now it's worth noting that the terminal side will end up in one of four quadrants. Those are these four regions bound by the x- and y-axes. We number these counterclockwise, quadrant one, two, three, and four. So an up-toos angle will end up in quadrant two. And if you remember that the measure of the angle is determined by the amount of rotation, if we keep rotating we could have the terminal side in quadrant three or quadrant four. So we say that the angle is in quadrant one, two, three, or four, depending on where the terminal side is located. Now when drawing angles in standard position it's helpful to remember that a full turn is the same as 360 degrees. And what that means is that a half turn is 180 degrees, a quarter turn is 90 degrees, and three quarters of a turn is 270 degrees. And what this means is that if you rotate by less than 90 degrees you're still in the first quadrant. And that's because you've made less than a quarter turn rotation. A rotation between 90 and 180 degrees puts you in the second quadrant. And that's because you've turned somewhere between a quarter of a turn and half of a turn. If your rotation is between 180 and 270 degrees you're in the third quadrant. And finally anything between 270 and 360 degrees puts you in the fourth quadrant. So for example let's draw an angle in standard position with a measure of 120 degrees, 225 degrees, and 330 degrees. So we might proceed as follows. If we do a quarter turn rotation that gets us 90 degrees. Since we've only measured out 90 degrees but want to measure out 120 we have to keep going. Now if we go all the way to a half turn that's 180 degrees which is too much. So we'll need to stop somewhere short of that half turn, somewhere in the second quadrant. And remember if it's not written down it didn't happen label the angle. 225 degrees, well that's more than a half turn but less than three quarters of a turn and so that's going to put us someplace here in the third quadrant. And if it's not written down it didn't happen. And 330 degrees, well that's someplace between three quarters of a turn and a full turn so that's going to place us here in the fourth quadrant. So another useful concept in all of this is known as the reference angle. Given an angle in standard position the reference angle has the same terminal side as one leg but the other leg is on the nearer of the positive or negative x-axis. Now ordinarily we worry about whether we have to do a clockwise or counterclockwise rotation and that'll give us a negative or positive sign for the angle. For the reference angle the measure of the reference angle is unsigned. We don't worry about whether it's positive or negative. So let's go back to those angles, 120, 225 and 330 and let's find the reference angle for these. So first we'll draw an angle with a measure of 120 degrees in standard position What are we missing? So the reference angle will be the angle where the terminal side is one leg and the other leg is the nearer of the positive or negative x-axis. And in this case the negative x-axis is closer so we'll draw that as the other leg of the angle. And so our reference angle will be the measure of this angle. Now a useful way of approaching finding the reference angle is to remember that a half turn is 180 degrees. And so we might begin as follows. Note that 120 is 180 minus 60 and so we can go to the terminal side by rotating a half turn counterclockwise, that's 180 degrees, and then rotating clockwise by 60 degrees. And although we used a clockwise rotation of 60 degrees the measure of the reference angle is unsigned so the reference angle is 60 degrees. We'll do the same thing for an angle of 225 degrees. We'll draw the angle in standard position. The terminal side is one leg. The other leg is the nearer of the positive or negative x-axis. That's going to be the negative x-axis. And again since our half turns are easy we might begin by noting that 225 is 180 plus 45 so we can get to the terminal side by rotating a half turn counterclockwise 180 degrees and then rotating an additional 45 degrees. And so the reference angle is 45 degrees. For 330 degrees, well since drawing a picture made finding a reference angle easy this time we won't draw a picture. Wait, what? No no no no no we want to do this the easy way. So again we'll start out by drawing an angle with a measure of 330 degrees in standard position. The terminal side is one leg. The other leg is the nearer of the positive or negative x-axis. And this time the positive x-axis is closer. Now it's useful to remember that a full turn is 360 degrees. And so we might note here that our measure of 330 is 360 minus 30 so we can get to the terminal side by rotating a full turn counterclockwise and then rotating clockwise by 30 degrees. And so our reference angle is 30 degrees. So how does this help us answer our original nonsense question? What is the sine, cosine, and tangent of 120 degrees? We'll take a look at that next.