6 2: Graphs of the Other Trigonometric Functions Mathematics LibreTexts

In this case, we add \(C\) and \(D\) to the general form of the tangent function. We can identify horizontal and vertical stretches and compressions using values of \(A\) Best etf to day trade and \(B\). The horizontal stretch can typically be determined from the period of the graph.

Analyzing the Graph of \(y = \cot x\)

Because there are no maximum or minimum values of a tangent function, the term amplitude cannot 11 best ways to invest $1000 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 \(A\). The cotangent function can be represented using more general mathematical functions. It is more useful to write the cotangent function as particular cases of one special function. That can be done using doubly periodic Jacobi elliptic functions that degenerate into the cotangent function when their second parameter is equal to or . Let’s modify the tangent curve by introducing vertical and horizontal stretching and shrinking.

Domain, Range, and Graph of Cotangent

1. (a) are the simple poles with residues .(b) is an essential singular point.
2. We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole.
3. As an example, let’s return to the scenario from the section opener.
4. The horizontal stretch can typically be determined from the period of the graph.
5. The cotangent function is used throughout mathematics, the exact sciences, and engineering.

Some functions (like Sine and Cosine) repeat foreverand are called Periodic Functions. So basically, if we know the value of the function from \(0\) to \(2\pi\) for the first 3 functions, we can find the value of the function at any value. More clearly, we can think of the functions as the values of a unit circle. Where contains the unit step, real part, imaginary part, the floor, and the round functions. In the points , where has zeros, the denominator of the last formula equals zero and has singularities (poles of the first order).

Also, we will see the process of graphing it in its domain. 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. Alternative names of cotangent are cotan and cotangent x. As we did for the tangent function, we will again refer to the constant \(| A |\) as the stretching factor, not the amplitude.

Example from before: 3 sin(100(t + 0. )

In the same way, we can calculate the cotangent of all angles of the unit circle. Access these online resources for additional instruction and practice with graphs of other trigonometric functions. The cotangent function is used throughout mathematics, the exact sciences, and engineering. The cotangent function is an old mathematical function. Euler (1748) used this function and its notation in their investigations. This means that the beam of light will have moved \(5\) ft after half the period.

Here is a graphic of the cotangent function for real values of its argument . The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by why bond prices and yields move in opposite directions the transformation to the original function’s input. Now that we can graph a tangent function that is stretched or compressed, we will add a vertical and/or horizontal (or phase) shift.

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. Thus, the graph of the cotangent function looks like this. 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.