Hyperbolic Functions cosh(2-3i)

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SUMMARY

The discussion focuses on calculating the hyperbolic function cosh(2-3i) using the formula for cosh(a±b). The correct expansion is derived as cosh(2)cos(3) - i sinh(2)sin(3). The participants emphasize the importance of understanding the relationships between hyperbolic and trigonometric functions, particularly the double angle formulas for both types of functions. This approach clarifies how to manipulate complex arguments in hyperbolic functions effectively.

PREREQUISITES
  • Understanding of hyperbolic functions, specifically cosh and sinh
  • Familiarity with complex numbers and their representation
  • Knowledge of trigonometric functions, particularly sin and cos
  • Ability to apply double angle formulas in mathematical expressions
NEXT STEPS
  • Study the derivation and applications of hyperbolic function identities
  • Learn about complex analysis and its relation to hyperbolic functions
  • Explore the properties of exponential functions in relation to trigonometric and hyperbolic functions
  • Practice solving problems involving hyperbolic functions with complex arguments
USEFUL FOR

Students studying complex analysis, mathematicians exploring hyperbolic functions, and anyone seeking to deepen their understanding of the relationship between hyperbolic and trigonometric functions.

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Homework Statement



Find the real part, the imaginary part, and the absolute value of:

cosh(2-3i)

Homework Equations





The Attempt at a Solution



I know how to write this using exponentials, but when I looked up the answer the book expanded cosh(2-3i) to cosh 2 cos 3 − i sinh 2 sin 3 , how in the world do you get there??

Thanks for the help
 
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You'll be happy with this answer, I believe.
If you remember from regular sin and cosine rules, you have the double angle formula:

sin(a+b) = sina*cosb + sinb*cosa.

Well, you have equivalent functions for hyperbolic functions:

sinh(a±b) = sinha*coshb ± sinhb*cosha

and

cosh(a±b) = cosha*coshb ± sinha*sinhb

You are going to use the second one. The ± means that if you use minus in the first ±, you use minus in the second part of the expression as well.
 

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