WEBVTT 00:00:06.358 --> 00:00:09.360 Hi. Today we're going to talk about the unit circle 00:00:09.362 --> 00:00:12.109 and how we can relate it to trigonometric functions. 00:00:12.111 --> 00:00:16.858 The unit circle is a circle of radius 1, and it has the equation 00:00:16.860 --> 00:00:21.371 x squared plus y squared equals 1. At least that's the simplest equation for it. 00:00:21.373 --> 00:00:30.622 Now we know that the circumference of the circle is equal to 2 pi r. 00:00:30.624 --> 00:00:37.872 But in the unit circle the radius is 1, so the circumference of a unit circle is 2 pi. 00:00:37.874 --> 00:00:41.872 This is going to help us when we define the trigonometric functions. 00:00:44.000 --> 00:00:48.645 Before we start defining the trigonometric functions using the unit circle, 00:00:48.647 --> 00:00:53.905 we're going to define the central angle to be t radians. 00:00:53.907 --> 00:00:58.155 Now you remember that you've learned already that the arc length, 00:00:58.157 --> 00:01:02.905 the length of any intercepted arc here, is equal to r times theta. 00:01:02.907 --> 00:01:10.168 But in the unit circle we have that r is equal to 1, and we're defining theta, 00:01:10.170 --> 00:01:16.920 the angle, to be t. So we have that s is equal to t. 00:01:16.922 --> 00:01:22.920 So in a unit circle, now this is not any circle, but in a unit circle 00:01:22.922 --> 00:01:28.170 the radian measure of the central angle is equal to the length of the intercepted arc. 00:01:30.418 --> 00:01:35.420 One thing I want to point out is that t can be any real number. 00:01:35.422 --> 00:01:39.900 You can go around this circle in either direction, positive or negative direction, 00:01:39.902 --> 00:01:42.420 and you can go around it as many times as you want. 00:01:42.422 --> 00:01:47.420 So we can define the trig functions using this because t is any real number. 00:01:47.422 --> 00:01:51.920 Now what you've already done in trig is you have defined, for general angles, 00:01:51.922 --> 00:01:55.670 we have defined the sine of theta to be the value y over r 00:01:55.672 --> 00:01:59.420 and the cosine of theta to be x over r. 00:01:59.422 --> 00:02:03.400 But in the unit circle, r is equal to 1. That's one of the nice things about it. 00:02:03.410 --> 00:02:06.920 It makes it so useful. So r is equal to 1, and now we can define 00:02:06.922 --> 00:02:13.983 the sine of t to just be the value y and the cosine of t to be the value x. 00:02:13.985 --> 00:02:17.484 Now remember this is just on the unit circle. This is not for anything. 00:02:19.485 --> 00:02:24.942 OK, so now we're able to define all six of the trigonometric functions 00:02:24.942 --> 00:02:29.190 using the unit circle. We have for any real number t, 00:02:29.192 --> 00:02:32.442 now remember it can be positive, it can be negative, it can be 0, 00:02:32.442 --> 00:02:36.942 and a point P (x,y) that corresponds to this real number t, 00:02:36.944 --> 00:02:39.942 we define our six trigonometric functions 00:02:39.944 --> 00:02:45.442 as the sine of t is equal to y, the cosine of t is equal to x, 00:02:45.444 --> 00:02:50.000 the tangent of t is equal to y over x but x cannot be zero. 00:02:50.002 --> 00:02:55.749 And then we have the reciprocal functions, cosecant of t is 1 over y, y not zero. 00:02:55.751 --> 00:03:00.500 The secant of t is 1 over x, this is the reciprocal, x not equal to 0, 00:03:00.502 --> 00:03:05.251 and the cotangent is the reciprocal of tangent, x over y with y not equal to zero. 00:03:05.253 --> 00:03:08.782 So you need to learn these six trigonometric functions 00:03:08.784 --> 00:03:10.541 as we proceed through the next examples. 00:03:12.291 --> 00:03:16.539 Now let's do an example. We're going to find the values of the 00:03:16.541 --> 00:03:21.792 trigonometric functions for t with the point negative 3/5, 4/5, 00:03:21.794 --> 00:03:25.290 which is on the unit circle. That's what's nice about the unit circle. 00:03:25.292 --> 00:03:29.542 It makes defining these functions very simple. We can write that 00:03:29.544 --> 00:03:34.041 the sine of t, now remember that we define the sine of t to be the value of y, 00:03:34.043 --> 00:03:39.300 so the sine of t is equal to 4/5. 00:03:39.302 --> 00:03:47.570 We define the cosine of t to be x, so the cosine of t is negative 3/5. 00:03:47.572 --> 00:03:57.570 The tangent is defined as y over x, and you might remember 00:03:57.572 --> 00:04:05.570 that's sine over cosine, same thing, and that would reduce to negative 4/3. 00:04:05.572 --> 00:04:09.320 And then you can find the reciprocal functions using the definitions. 00:04:11.070 --> 00:04:15.571 Anytime you're working with functions you need to understand and be aware 00:04:15.573 --> 00:04:19.070 of what the domain and range of the function is. 00:04:19.072 --> 00:04:25.115 We define the sine of t to be y, and we already talked about the fact that 00:04:25.117 --> 00:04:28.364 t can be any real number. You can go around the unit circle. 00:04:28.366 --> 00:04:30.864 You can go around in the positive direction, negative direction, 00:04:30.866 --> 00:04:32.864 so t can be any real number. 00:04:32.866 --> 00:04:36.127 So the domain which is the input for the function, that's what you can put 00:04:36.129 --> 00:04:40.875 into the function, is any real number. So the domain here is going to be all reals 00:04:40.877 --> 00:04:43.381 which we will just designate with the interval notation 00:04:43.383 --> 00:04:46.630 negative infinity to infinity. That's all reals. 00:04:46.632 --> 00:04:50.600 Now the range of the function, that's what comes out of the function. 00:04:50.602 --> 00:04:53.144 That's the number that the function can actually equal. 00:04:53.146 --> 00:04:56.549 And if you look at the unit circle, and in general this is true 00:04:56.549 --> 00:04:58.894 for all sine and cosine functions, 00:04:58.896 --> 00:05:03.390 the value that the sine function is going to take on, the value that you put 00:05:03.392 --> 00:05:05.896 a number in, you take the sine, you get a number out, 00:05:05.898 --> 00:05:10.143 that number is always going to be between one and negative one. 00:05:10.145 --> 00:05:17.144 So the range of the sine function is the interval negative one to one, 00:05:17.146 --> 00:05:19.920 and the same is going to be true for the cosine function. 00:05:19.922 --> 00:05:23.671 So the domain for the sine and the cosine function is all reals. 00:05:23.673 --> 00:05:27.428 The range for the sine and the cosine function is always 00:05:27.430 --> 00:05:31.180 between negative one and one inclusive.