 So let's take a look at trigonometric functions beyond the right angle. So let's consider this trigonometry in the unit circle. Suppose our angle has been moved into standard position. So remember that's going to have the vertex at the origin, one side on the positive x-axis, and other side wherever it ends up. Now let's consider a unit circle centered at the origin. That's a circle with radius one centered at the origin. The terminal side of the angle will intersect the unit circle at some point b. And since we're on a set of coordinate axes, we can say that this point b has coordinates x, y. Now let's consider this coordinate a little bit more carefully. If the point b has coordinates x, y, then this vertical line, b, c, has length y. Likewise, this horizontal line, a, c, has length x. And a useful rule in mathematics is look for the right triangle. Since b, c is vertical, that means that this angle b, c, a is a right angle, and so a, b, c is a right triangle. Now I know a, c is x and b, c is y, but what about this third side, a, b? Well remember, that's the distance between the center of the circle and the point of the circle, and since this is a unit circle, the radius a, b has length one. Now since a, b, c is a right triangle, we can talk about the sine, cosine, and tangent of this angle a. So the sine of a is going to be the length of the opposite side divided by the length of the hypotenuse. That opposite side has length y, and the hypotenuse, because it's the radius of the unit circle, has length one. So the sine is y over one, or just y. Similarly, the cosine of the angle is the adjacent over the hypotenuse, and that's going to be x over one, or just x. And finally, the tangent is going to be the opposite over the adjacent, and that's going to be y over x. So now let's change our viewpoint slightly. Suppose angle a is in standard position. Let b be the point where the terminal side intersects the unit circle with coordinates x, y. Then the sine of a is equal to y, the cosine of a is equal to x, and the tangent of a is equal to y over x. And what's useful to keep in mind here is while this was determined by assuming that this angle a was an acute angle, we can still find these values, x, y, and y over x, whatever the measure of the angle a is. And this allows us to introduce the following. Let theta be an angle in standard position and let its terminal side intersect the unit circle centered at the origin at a point with coordinates x, y. Then the sine of theta is y, the cosine of theta is x, and the tangent of theta is y over x. So for example, suppose we have the circle shown, let's find the sine, cosine, and tangent for the angle. So let's pull in our definition. So sine is equal to the y-coordinate, cosine is equal to the x-coordinate, and tangent is the quotient of the y and x-coordinates. Now in a kind and gentle universe, we'd have the x and y-coordinates. Unfortunately, the universe isn't so kind and generous. What's that? Oh, here we have the x and y-coordinates. Since sine is our y-coordinate, our sine is going to be 513s. Since cosine is the x-coordinate, our cosine will be minus 1213s. And since tangent is the quotient of the y over the x-coordinates, it's going to be negative 512s. Now we have to find sine, cosine, and tangent for any angle theta. But the other trigonometric functions can also be defined in terms of sine, cosine, and tangent. And so this allows us to find the rest for any angle theta. secant theta is 1 over cosine, cosecant is 1 over sine, and cotangent is 1 over tangent. So we can try to find this secant, cosecant, and cotangent for the angle shown. Now we've already found the sine, cosine, and tangent, and the thing to remember is the secant, cosecant, and cotangent are the reciprocals of these functions. So secant is the reciprocal of cosine, so it's going to be... cosecant is the reciprocal of sine, and cotangent is the reciprocal of tangent. So these will be...