Cotangent: Introduction to the Cotangent Function

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 table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians. The cotangent function can be represented using more general mathematical functions. As the ratio of the cosine and sine functions that are particular cases of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the cotangent function can also be represented as ratios of those special functions. It is more useful to write the cotangent function as particular cases of one special function.

  • 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.
  • The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator.
  • As usual, the inverse trigonometric functions are denoted with the prefix “arc” before the name or its abbreviation of the function.
  • In geometric applications, the argument of a trigonometric function is generally the measure of an angle.

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real). 🔎 You can read more about special right triangles by using our special right triangles calculator. Hypothetical performance results have many inherent limitations, some of which are
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described below.

In the same way, we can calculate the cotangent of all angles of the unit circle. 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. The cotangent of an angle in a right triangle is defined as the ratio of the adjacent side (the side adjacent to the angle) to the opposite side (the side opposite to the angle). Here is a graphic of the cotangent function for real values of its argument . We can identify horizontal and vertical stretches and compressions using values of \(A\) and \(B\).

In this section, let us see how we can find the domain and range of the cotangent function. The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known. It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

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. 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. Needless to say, such an angle can be larger than 90 degrees.

3: Graphs of the Tangent and Cotangent Functions

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.

The sine and cosine functions are one-dimensional projections of uniform circular motion. Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. However, on each interval on which a trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function.

Various mnemonics can be used to remember these definitions. 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. 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. However, let’s look closer at the cot trig function which is our focus point here.

Sign of Cotangent

The beam of light would repeat the distance at regular intervals. 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.

How To Use COT In Day Trading?

Such simple expressions generally do not exist for other angles which are rational multiples of a right angle. 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.

What is the period for csc, sec, and cot?

This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem. This formula is commonly considered for real values of x, but it remains true for all complex values. In geometric applications, the argument of a trigonometric function is generally the measure of an angle. One common unit is degrees, in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics). Together with the cot definition from the first section, we now have four different answers to the “What is the cotangent?” question. It seems more than enough to leave the theory for a bit and move on to an example that actually has numbers in it.

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.

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 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.

Cotangent in Terms of Cos and Sin

As with the sine and cosine functions, the tangent function can be described by a general equation. In this section A, B, C denote the three (interior) angles of a triangle, and a, b, c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve. how to buy polygon crypto The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler’s formula.

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Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions. This is the tangent half-angle substitution, which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions. Trigonometric functions are differentiable and analytic fibonacci extension levels at every point where they are defined; that is, everywhere for the sine and the cosine, and, for the tangent, everywhere except at π/2 + kπ for every integer k. 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.

Tangent and Cotangent Graphs

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performance is not necessarily indicative of future results. In the points , where has zeros, the volatility trading strategies denominator of the last formula equals zero and has singularities (poles of the first order). All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above. When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

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