Cosech, also known as hyperbolic cosecant, is a mathematical function that is used in hyperbolic trigonometry. It is the reciprocal of the hyperbolic sine function (sinh) and is defined as the ratio of the hypotenuse to the opposite side in a right triangle.
The study of hyperbolic functions, including cosech, can be traced back to the 18th century. Swiss mathematician Leonhard Euler is credited with introducing these functions and their properties. Since then, cosech has been extensively used in various branches of mathematics, physics, and engineering.
Cosech is typically introduced in advanced high school mathematics or college-level courses. It is a topic covered in trigonometry or calculus classes.
To understand cosech, it is important to have a solid foundation in trigonometry and exponential functions. Here is a step-by-step explanation of cosech:
Cosech is defined as the reciprocal of the hyperbolic sine function (sinh). It can be expressed as:
cosech(x) = 1 / sinh(x)
The hyperbolic sine function (sinh) is defined as:
sinh(x) = (e^x - e^(-x)) / 2
where e is the base of the natural logarithm.
By substituting the definition of sinh into the cosech equation, we get:
cosech(x) = 1 / ((e^x - e^(-x)) / 2)
Simplifying further, we can rewrite cosech as:
cosech(x) = 2 / (e^x - e^(-x))
Cosech is a single-valued function that is defined for all real numbers except x = 0. It is an odd function, which means that cosech(-x) = -cosech(x) for any value of x.
Some important properties of cosech include:
The range of cosech is (-∞, -1] ∪ [1, ∞).
Cosech is an odd function, as mentioned earlier.
Cosech is periodic with a period of 2πi, where i is the imaginary unit.
The derivative of cosech with respect to x is -cosech(x) * coth(x).
To find the value of cosech for a given angle or real number, you can use a scientific calculator or computer software that has built-in trigonometric functions. Simply input the angle or number and select the cosech function to obtain the result.
The formula for cosech is:
cosech(x) = 2 / (e^x - e^(-x))
The cosech formula can be applied in various mathematical and scientific contexts. It is commonly used in solving differential equations, analyzing waveforms, and studying the behavior of physical systems.
For example, in physics, the cosech function can be used to describe the shape of a vibrating string or the decay of radioactive materials.
The symbol or abbreviation for cosech is "csch".
There are several methods for calculating cosech, including using the power series expansion, numerical approximation methods, or utilizing the properties and identities of hyperbolic functions.
Example 1: Find the value of cosech(2).
Solution: Using the formula cosech(x) = 2 / (e^x - e^(-x)), we can substitute x = 2:
cosech(2) = 2 / (e^2 - e^(-2))
Using a calculator, we find that cosech(2) ≈ 1.099.
Example 2: Solve the equation cosech(x) = 3.
Solution: Rearranging the formula, we have:
3 = 2 / (e^x - e^(-x))
Multiplying both sides by (e^x - e^(-x)), we get:
3(e^x - e^(-x)) = 2
Expanding and rearranging, we have:
3e^x - 3e^(-x) = 2
Letting y = e^x, we can rewrite the equation as:
3y - 3/y = 2
Simplifying further, we get a quadratic equation:
3y^2 - 2y - 3 = 0
Solving this quadratic equation, we find two possible values for y. Substituting back to find x, we obtain the solutions.
Example 3: Evaluate the integral ∫ cosech(x) dx.
Solution: Using the property that the integral of cosech(x) is equal to the natural logarithm of the absolute value of cosech(x) plus the hyperbolic tangent of x, we have:
∫ cosech(x) dx = ln|cosech(x)| + tanh(x) + C
where C is the constant of integration.
Find the value of cosech(0).
Solve the equation cosech(x) = 0.
Evaluate the integral ∫ cosech^2(x) dx.
Q: What is the relationship between cosech and sinh? A: Cosech is the reciprocal of sinh. In other words, cosech(x) = 1 / sinh(x).