For example, for k = 0, we have the cot 0 that we've been so worried about. That is precisely what happens with cot x in points x = k♱80°. As we all know, we can't have zero in the denominator. After all, according to the above section and the cotangent formula we had there, the map is defined as a fraction. The fact that the function is undefined in some places (for instance, cot 0 is undefined) results from the cot definition. Let's take a moment and add a few more words to the last point above. On the cotangent graph above, we see that the curve goes to plus/minus infinity at these points. For angles x of the form x = k♱80° with k an integer, cot x is undefined. Similarly to the tangent (and, in fact, the secant and cosecant), the cotangent function doesn't always exist. In mathematical notation, we can write this fact as cot(x) = cot(x + 360°). This characteristic means that the function's values repeat every 360 degrees. The cotangent function is periodic with a 360-degree period. In other words, we have cot(x) = -cot(-x). This means that the value at angle x is the opposite of that at -x. This means that for some angles, it will be tiny (say, -10,000,000 or -10⁷ if you prefer scientific notation), while for others, it'll be quite large.
The cotangent function admits all real values. To have it all neat in one place, we listed them below, one after the other. We can already read off a few important properties of the cot trig function from this relatively simple picture. And since we like pretty pictures, we'll start by drawing the cotangent graph. We've established the cot definition that we're all happy with (we are, aren't we?), so it's time for the next step: analyzing the cot function. For instance, what would cot 0 be? After all, for such an angle, the y coordinate is zero, and we can't divide by zero, can we?
This way, we get a new cotangent formula:īut there are new questions to answer. So what is cot in this new language? In the trigonometric function definitions above, we substitute a for y, b for x, and c for √(x² + y²) (the distance from (0,0) to A that corresponds to the hypotenuse). What is more, since we've directed α, we can now have negative angles as well by simply going the other way around, i.e., clockwise instead of counterclockwise.
#TANGENT GRAPH FULL#
For that, we just consider 360 to be a full circle around the point (0,0), and from that value, we begin another lap. We can even have values larger than the full 360 -degree angle. Needless to say, such an angle can be larger than 90 degrees.
Because of that, we often call α a directed angle.) (Observe how we said that α goes from one line to the other and not that it is between them. Let A = (x,y) be a point on the plane, and let α be the angle going counterclockwise from the positive half of the horizontal axis to the line segment connecting (0,0) and A. The trick is to move the whole reasoning to the two-dimensional Euclidean space, i.e., the plane. Fortunately for us, there is a way to have all of the functions (including our beloved cot trig function) extended to any angle, even a negative one. So know this graph because in a future episode, we're going to extend this in both directions because tangent's actually defined for all real numbers.With all their strengths, there is also a slight weakness to the cot definition we gave above: it only allows angles from 0 to 90 degrees (or from 0 to π/2 in radians).
It's approaching infinity and that's why the tangent zooms off to infinity. As this angle gets closer and closer to pi over 2, the slope of this line gets steeper and steeper. And the reason for that is again it comes back to slope. It just increases steep more steeply and steeply as x approaches or as theta rather approaches pi over 2. It increases very rapidly like that and it actually has a vertical asymptote. If that's 1.5 and that's 2, 1.7 is about here. Root 3 is approximately 1.7, so I'm going to plot that as 1.7, and pi over 3 is two thirds the way from 0 to pi over 2. Pi over 4 is halfway between 0 and pi over 2, so right here. And I'm just going to use these 2 points. The first point is 0 0, that goes right there. Let's start by plotting some points, I'll come back to the slope issue in a second. But tangent gives me the slope of this line.Īlright. So y over x is the slope of op and that kind of helps us see how tangent behaves. And the slope of this line would be y over x rise over run. Why is that? Well it's because you draw this little triangle here, the vertical leg of the triangle is y and the horizontal leg is x where x and y are these coordinates. I want to remind you that another way to see the tangent function as the slope of the terminal side op. I have a table of values written here and the definition of the tangent function on the unit circle here.
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