Welcome to Omni's cotangent calculator, where we'll study the cot trig function and its properties. Arguably, among all the trigonometric functions, it is not the most famous or the most used. Nevertheless, you can still come across cot x (or cot(x)) in textbooks, so it might be useful to learn how to find the cotangent. Fortunately, you have Omni to provide just that, together with the cot definition, formula, and the cotangent graph.

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So what is this cot exactly? Well, why don't we jump to the first section and find out?


What is cot x? The cotangent definition

Quite possibly (although we can never be sure), when ancient Greeks began studying triangles, they were not aware of what they started. For instance, when Pythagoras came up with his theorem, he probably didn't think to himself: "I bet it'll lead to some flimsy curves that all high school students will have to memorize someday." Still, that is precisely what happened.

Even if their attempts at looking into the future were faulty at best, they did get one thing right: right triangles are important. It turned out that not only do they have to satisfy the famous a² + b² = c² formula but also their sides and inside angles are connected. After all, we can easily imagine that if one angle (not the right one, mind you) increases, then the opposite side must get larger as well. This concept, in essence, is the idea behind trigonometry.

Trigonometric functions describe the ratios between the lengths of a right triangle's sides. Below you can find them all, including the cot definition.

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So what is the cotangent? According to the picture above, the cot formula is: divide the side next to the angle by the other one. What is more, the strength of that definition (and the other five, in fact) lies in the fact that, in some sense, the lengths of the triangle's sides do not matter. That is, if we scale the triangle to a larger or smaller size but keep the angle intact, the ratios (and with them, the cotangent function and the others) will not change.

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). After all, it is a right triangle. 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. The trick is to move the whole reasoning to the two-dimensional Euclidean space, i.e., the plane.

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. (Observe how we said that α goes from one line to the other and not that it is between them. Because of that, we often call α a directed angle.)

Needless to say, such an angle can be larger than 90 degrees. We can even have values larger than the full 360-degree angle. For that, we just consider 360 to be a full circle around the point (0,0), and from that value, we begin another lap. 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.

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). This way, we get a new cotangent formula:

cot(α) = x / y.

But there are new questions to answer. 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?

Still, we got some answers so far. 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. And since we like pretty pictures, we'll start by drawing the cotangent graph.


Cot graph

You might have already seen the graph of the tangent function. Believe it or not, the similarity in the name is not coincidental. Who'd have thought, right? So if you recall the tangent function graph, you can play "spot the difference" with the cotangent graph below. (Note that this is the same picture that we use in our cotangent calculator.)

Observe that this is quite a special triangle in which we know the relations between the sides, i.e., we can be sure that if the shorter leg is of length x, then the hypotenuse will be 2x. This is because our shape is, in fact, half of an equilateral triangle. As such, we have the other acute angle equal to 60°, so we can use the same picture for that case.

Recall that cot in math is the ratio of the leg next to the angle to the other one. So what is the cotangent in this case? We have:

cot(30°) = x√3 / x = √3,

and:

cot(60°) = x / x√3 = 1 / √3 = √3 / 3.

Next, we move on to the 45° angle.

Again, we are fortunate enough to know the relations between the triangle's sides. This time, it is because the shape is, in fact, half of a square.

We use the cotangent definition to get:

cot(45°) = x / x = 1.

So we're left with the last angle. What is the cotangent of 75°?

Well, as it occurs, the answer is not so simple. We have no special triangle to use here. We could, for instance, recall the previous section and find the answer by calculating tan(75°) first. For that, however, we would have to use, for instance, the half-angle formulas, which would, in turn, require us to find cos(150°). This problem, eventually, is not that difficult since 150° = 180° - 30°, and both 30° and 180° are fairly simple angles.

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The lesson here is that in general, calculating trigonometric functions is no walk in the park. In fact, we usually use external tools for that, such as Omni's cotangent calculator.

Nevertheless, these few simple calculations were surely a good preparation for the upcoming test. Once you get your final grade in mathematics, look back at all the memories you've shared with Omni Calculator that helped you along the way, and give us a satisfied nod of the head. The poor content developers are hungry for those. 😀