Question

Prove that $\cos z=\cos x \cosh y-i \sin x \sinh y$

   Prove that $\cos z=\cos x \cosh y-i \sin x \sinh y$

Advanced Engineering Mathematics

Advanced Engineering Mathematics

Dennis G. Zill,… 5th Edition

Chapter 17, Problem 23

Instant Answer

Solved by Expert Varsha Aggarwal

01/07/2022

Step 1

Step 1: We start with the formula for the cosine of a complex number, which is given by: \[ \cos z = \frac{e^{iz} + e^{-iz}}{2} \] Show more…

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Prove that $\cos z=\cos x \cosh y-i \sin x \sinh y$

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Key Concepts

Euler's Formula

Euler's formula, which states that e^(i?) = cos ? + i sin ?, is a cornerstone in complex analysis. It bridges the gap between exponential functions and trigonometric functions, enabling the extension of trigonometric identities into the complex domain.

Exponential Representation of Trigonometric Functions

Trigonometric functions, such as cosine, can be represented using complex exponentials; for example, cos z is expressed as (e^(iz) + e^(-iz)) / 2. This representation is particularly useful when dealing with complex numbers, as it simplifies the handling of real and imaginary parts.

Hyperbolic Functions

Hyperbolic functions like cosh y and sinh y are defined by exponential formulas: cosh y = (e^y + e^(-y)) / 2 and sinh y = (e^y - e^(-y)) / 2. They appear naturally when extending trigonometric identities to complex arguments, especially under the substitution of an imaginary unit.

Separation of Real and Imaginary Parts

In complex analysis, it is often necessary to separate a function into its real and imaginary components. By expressing complex functions in this way, one can compare corresponding parts to verify identities or simplify expressions, as demonstrated in the proof of complex trigonometric identities.

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