Compute the real and imaginary parts: (a) e^2+i (b) sin(1+i) (c) e^3-i (d) cos(2+3 i)

Name: Problem 6, Chapter 2: Linear Algebra
Uploaded: 2020-07-29T20:16:33-08:00
Duration: 4 min 6 s
Channel: Rakesh Kumar Sharma
Description: Compute the real and imaginary parts: (a) e^2+i (b) sin(1+i) (c) e^3-i (d) cos(2+3 i)

step 1 In this question we have to simplify the expression that is given and we have to find out the re

Compute the real and imaginary parts: (a) e^2+i (b) sin(1+i)...

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Key Concepts

Hyperbolic Functions

Hyperbolic functions, including cosh and sinh, are analogous to the trigonometric functions but for a hyperbola. They are defined in terms of exponentials and appear naturally in the expressions for sine and cosine when extended to complex arguments, aiding in the computation of real and imaginary parts.

Euler's Formula

Euler's Formula, e^(i?) = cos(?) + i sin(?), is fundamental in complex analysis. It relates complex exponentials to trigonometric functions and is used to decompose complex exponential expressions into their real and imaginary parts.

Exponential Function for Complex Numbers

When dealing with the exponential of a complex number, such as e^(a+bi), it is standard to rewrite it as e^a * e^(bi). This representation leverages Euler's Formula, allowing the separation into e^a [cos(b) + i sin(b)], which directly provides the real and imaginary parts of the expression.

Trigonometric Functions with Complex Arguments

The sine and cosine functions can be extended to complex arguments using definitions that involve hyperbolic functions. For example, sin(a+bi) and cos(a+bi) are expressed in terms of combinations of sine, cosine, hyperbolic sine (sinh), and hyperbolic cosine (cosh), enabling the extraction of their real and imaginary components.

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