Manipulating hyperbolic functions

In summary, the problem asks for the function cosh(6x) to be expressed in terms of powers of cosh(x). To solve this, we can use the double angle formula for cosh(a+b) and break down 6x into 3x+3x. From there, we can use multiple angle formulas and work our way down to powers of cosh(x).
  • #1
pokgai
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Homework Statement


Express the function cosh(6x) in terms of powers of cosh(x)

Homework Equations


The Attempt at a Solution


Okay the problem booklet also asks me to do the opposite. Express cosh(x)^6 as mutiples of cosh(x). I can do that fine, I just simply write it out as [1/2(e^x + e^-x)]^6 etc. and then expand using pascals and group the like terms and that gives me multiples.

However I have no idea where to begin when I'm doing the opposite. I assume I use double angle formulas as I have nothing else? hints would be appreciated

Cheers
 
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  • #2
cosh(a+b)=cosh(a)*cosh(b)+sinh(a)*sinh(b). 6x=3x+3x. Now work your way down to powers of cosh(x). Yes, there are multiple angle formulas. There's a similar one for sinh.
 

1. What are hyperbolic functions?

Hyperbolic functions are a set of mathematical functions that are similar to trigonometric functions, but are based on the hyperbola instead of the circle. These functions are widely used in mathematics, physics, and engineering.

2. How do hyperbolic functions differ from trigonometric functions?

Hyperbolic functions use the hyperbola as their base shape, while trigonometric functions use the circle. This results in different values and properties for the two sets of functions. For example, the hyperbolic sine function (sinh) is always positive, while the sine function (sin) can be both positive and negative.

3. What are the most commonly used hyperbolic functions?

The most commonly used hyperbolic functions are the hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh). These functions have many applications in areas such as calculus, differential equations, and physics.

4. How are hyperbolic functions manipulated in calculus?

Hyperbolic functions can be manipulated using similar techniques to trigonometric functions in calculus. For example, the chain rule and product rule can be applied to find derivatives of hyperbolic functions. Additionally, integration techniques such as substitution and integration by parts can be used to integrate hyperbolic functions.

5. What are some real-world applications of hyperbolic functions?

Hyperbolic functions have many practical applications, including modeling the shape of a hanging cable under tension, calculating the trajectory of a projectile, and describing the behavior of electric currents in circuits. They are also used in fields such as architecture, economics, and statistics.

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