Graphs of sech(x), cosech(x) and coth(x)

Understanding the Graphs of sech(x), cosech(x) and coth(x)

Basic Definitions

• sech(x) stands for the hyperbolic secant of x, which is defined as 1/cosh(x).
• cosech(x) stands for the hyperbolic cosecant of x, which is calculated as 1/sinh(x).
• coth(x) stands for the hyperbolic cotangent of x, defined as cosh(x)/sinh(x).

Graph Characteristics

• Graphs of sech(x) and cosech(x) approach zero as x goes to plus or minus infinity.
• The sech(x) graph is an even symmetric curve, appearing as a series of arches above the x-axis.
• The cosech(x) graph has two branches, one in each quadrant. The branches of this curve approach zero and are asymptotic to the x-axis.
• The coth(x) graph also features two branches: one in each hemisphere, each tending towards the lines y=1 and y=-1.

Key Features

• The hyperbolic secant and hyperbolic cosecant, sech(x) and cosech(x), are reciprocals of hyperbolic cosine and hyperbolic sine respectively.
• The hyperbolic cotangent, coth(x), is the reciprocal of the hyperbolic tangent.
• Asymptotes are an important feature in the graphs of cosech(x) and coth(x), which are not present in the graph of sech(x).

Domain and Range

• sech(x) and cosech(x) have a domain of all real numbers, but their range is restricted to positive values only.
• coth(x) has a domain of all real numbers except zero, with no restrictions on its range.

Transformations and Translations

• Like other functions, sech(x), cosech(x), and coth(x) can be subject to transformations, such as scaling, reflection, and translation.
• For instance, translating the graph of sech(x) upwards by 1 unit results in the graph of sech(x) + 1.

Practical Applications

• These functions have real-life applications in mechanical and electrical engineering, especially in wave form analysis and in the modelling of certain types of oscillators.