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Chapter 2: Q20P (page 74)

Evaluate each of the following in x+ iyform, and compare with a computer solution.

sin(iln1-i1+i)

Short Answer

Expert verified

The value of siniln1-i1+iis, 1.

Step by step solution

01

Given Information.

The given number is, siniln1-i1+i.

02

Definition of Complex Number.

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

03

Find the value of sin(i ln1-i1+i) .

Let the complex number as.u=siniln1-i1+i

Use the identity sin(iz)=isinh(z) .

u=isinhln1-i1+i

Let,z1=1-i

Find the argument and the angle of z1=1-i.

localid="1658902575018" r1=1-i=2胃1=arctan-1=-蟺4

The angle is in4thquadrant; therefore it is valid for using in the solution.

Let,z2=1+i

r2=1-i=2胃2=arctan1=蟺4

The angle is in1st quadrant therefore it is valid for using in the solution.

u=isinhIn2exp-蟺/42exp蟺/4=isinhInexp-蟺颈2=isinhIn1-蟺颈2卤2苍颈蟺=-sin-蟺颈2卤2苍颈蟺=1n=0,1,2,3,...

Hence, the value ofsiniln1-i1+iis 1 .

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Most popular questions from this chapter

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

Rez=0.

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

The anglez=蟺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鈥攖ry 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.

12.4e-8i蟺/3

Express the following complex numbers in the 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鈥攖ry 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.

17.11+i3

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find x,y,r,胃in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

role="math" localid="1658476746206" (cos3蟺2+isin3蟺2)
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