Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let’s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? Just like other trigonometric ratios, the cotangent formula is also defined as the ratio of the sides of a right-angled triangle.
Since the cotangent function is NOT defined for integer multiples of π, there are vertical asymptotes at all multiples of π in the graph of cotangent. Also, from the unit circle (in one of the previous sections), we can see that cotangent is 0 at all odd multiples of π/2. Also, from the unit circle, we can see that in an interval say (0, π), the values of cot decrease as the angles increase.
As with the sine and cosine functions, the tangent function can be described by a general equation. In the same way, we can calculate the cotangent of all angles of the unit circle. If in a triangle, we know the adjacent and opposite sides of an angle, then by finding the inverse cotangent function, i.e., cot-1(adjacent/opposite), we can find the angle. In this section, let us see how we can find the domain and range of the cotangent function. We can identify horizontal and vertical stretches and compressions using values of \(A\) and \(B\).
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.
Sign of Cotangent
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. 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. Together with the cot definition from the first section, we now have four different answers to the “What is the cotangent?” question.
- In this case, we add \(C\) and \(D\) to the general form of the tangent function.
- As with the sine and cosine functions, the tangent function can be described by a general equation.
- It seems more than enough to leave the theory for a bit and move on to an example that actually has numbers in it.
- However, let’s look closer at the cot trig function which is our focus point here.
In fact, we usually use external tools for that, such as Omni’s cotangent calculator. Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant \(A\). Let’s modify the tangent curve by introducing vertical and horizontal stretching and shrinking.
What is Cotangent in Trigonometry?
The cot x formula is equal to the ratio of the base and perpendicular of a right-angled triangle. Here are 6 basic trigonometric functions and their abbreviations. From the graphs of the tangent and cotangent functions, we see that the period of tangent and cotangent are both \(\pi\). In trigonometric identities, we will see how to prove the periodicity of these functions using trigonometric identities. It is, in fact, one of the reciprocal trigonometric ratios csc, sec, and cot.
It is usually denoted as “cot x”, where x is the angle between the base and hypotenuse of a right-angled triangle. Let us learn more about cotangent by learning its definition, cot x formula, its domain, range, graph, derivative, and integral. Also, we will see what are the values of cotangent on a unit circle. The lesson here is that, in general, calculating trigonometric functions is no walk in the park.
It seems more than enough to leave the theory for a bit and move on to an example that actually has numbers in it. However, let’s look closer at the cot trig function which is our focus point here. This means that the beam of light will have moved \(5\) ft after half the period. Again, we are fortunate enough to know the relations between the triangle’s sides.
Cotangent in Terms of Cos and Sin
This time, it is because the shape is, in fact, half of a square. Also, observe how for 30° and 60°, it gives you precise values before rounding them up, i.e., in the form of a fraction with square roots. 🙋 Learn more about the secant function with our secant calculator.
What are the Derivative and Integral of Cot x?
We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a police car parked next to a warehouse. The rotating https://www.dowjonesrisk.com/ light from the police car would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals.
We can determine whether tangent is an odd or even function by using the definition of tangent. Note, however, that this does not mean that it’s the inverse function to the tangent. That would be the arctan map, which takes the value that the tan function admits and returns the angle which corresponds to it. Here, we can only say that cot x is the inverse (not the inverse function, mind you!) of tan x. Needless to say, such an angle can be larger than 90 degrees. We can even have values larger than the full 360-degree angle.
Now that we can graph a tangent function that is stretched or compressed, we will add a vertical and/or horizontal (or phase) shift. In this case, we add \(C\) and \(D\) to the general form of the tangent function. They announced a test on the definitions and formulas for the functions coming later this week. We can already read off a few important properties of the cot trig function from this relatively simple picture. To have it all neat in one place, we listed them below, one after the other. Trigonometric functions describe the ratios between the lengths of a right triangle’s sides.
The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and cotangent functions.
The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph. 🔎 You can read more about special right triangles by using our special right triangles calculator. In fact, you might have seen a similar but reversed identity for the tangent. If so, in light of the previous cotangent formula, this one should come as no surprise.
🙋 Give our trigonometry calculator a try if you want to discover more about the trigonometry world.