You are here:

# Calculus/Hyperbolic functions

Question
Hi, I'm currently having problems answering some questions on deriving hyperbolic statements from trigonometric identities. A question is to derive the hyperbolic statement from sin^2 (x) + cos^2 (x). Any help on how to do this would be greatly appreciated. Thanks in advance

Questioner:Max
Country:Sheffield, United Kingdom
Category:Calculus
Private:No

Subject:Hyperbolic functions

Question:
Hi, I'm currently having problems answering some questions on deriving hyperbolic statements from trigonometric identities. A question is to derive the hyperbolic statement from sin^2 (x) + cos^2 (x). Any help on how to do this would be greatly appreciated. Thanks in advance

...............................
I don't know what you mean by "deriving hyperbolic statements".  It may be that your teacher ... oops, I mean your tutor  (I forgot -- you don't have teachers.) means to derive an identity involving hyperbolic functions, based, I assume, on their definitions.

And so, there is a hyperbolic identity similar to sin^2 (x) + cos^2 (x) = 1. I think it is something like cosh^2(x) - sinh^2(x) = 1.
......................
https://en.wikipedia.org/wiki/Hyperbolic_function#Useful_relations

has a few of them.
......................
You would derive that right from the definitions; just do some algebra.
========================================================================
Followup:

As a useful way to exercise your algebraic skills:

Starting with the definition:

cosh(x) = 1/2(exp(x) + exp(-x))

derive the formula for the INVERSE hyperbolic cosine:

arccosh(x).

1. Write  y = arccosh(x), which means:

2. x = cosh(y)

3. Write the definition for cosh(y)

4. For convenience, write Y = exp(y).

5. Solve the equation for Y in terms of x. (It's a quadratic -- use the formula)

6. Finally get:

arccosh(x) = ln(x + sqrt(x^2 - 1))

Calculus

Volunteer

#### Paul Klarreich

##### Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

##### Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)