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Chapter 2: Q4MP (page 80)

Find one or more values of each of the following complex expressions and compare with a computer solution.

(1+πi2)

Short Answer

Expert verified

The value of sinh1+πi2is, 1.54i .

Step by step solution

01

Given Information.

The given hyperbolic number is1+πi2 .

02

Definition of Complex Number.

The numbers that are presented in the form ofx+iy where, ' x ' is real numbers and ' i y ' is an imaginary number, those numbers are referred to as called Complex numbers.

03

Find the value of sinh(1+πi2) .

Use the following mentioned identities for solving the question.

sinh(yi)=isin(y)cosh(yi)=cos(y)sinh(x+yi)=sinh(x)cosh(yi)+cosh(x)sinh(yi)=sinh(x)cos(y)+icosh(x)sin(y)......(1)

Substitute,

x=1y=π2

in equation (1).

sinh1+π2i=sinhcosπ2+icosh(1)sinπ2=icosh(1)=1.54i

Hence, the value ofsinh1+π2i is1.54i .

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Most popular questions from this chapter

Express the following complex numbers in the x+iyform. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

9.2e-iπ/2

Express the following complex numbers in thex+iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

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Question: Describe geometrically the set of points in the complex plane satisfying the following equations.

z2=-z¯2..

Express the following complex numbers in the x+iyform. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

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Question:First simplify each of the following numbers to the x+iy form or to the reiθform. Then plot the number in the complex plane.

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