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Chapter 2: Q30P (page 71)

Find each of the following in the x+iy form and check your answers by computer.

tanh(3πi4)

Short Answer

Expert verified

tanh3πi4Can be written in the form of0-i=i .

Step by step solution

01

Given Information.

The given expression istanh3πi4 .

02

Definition of Complex Numberand Hyperbolic functions.

The domain of a function refers to the range of values that can be plugged into it. This is the set of x values in a function like f (x). The range of a function is the set of values that the function can take. This is the set of values that the function produces when we enter x value. The y values are what you're looking for.

The basic hyperbolic functions are:

  1. Hyperbolic sine(sinh)
  2. Hyperbolic cosine(cosh)
  3. Hyperbolic tangent(tanh)
03

Writetanh(3xi4) in the form of x+iy.

Let u=tanh3πi4

It is known that tanhzi=itanz and

tan3π4=-1

u=itan3π4=i-1=-i

tan3π4=-1

Hence,tanh3πi4 can be written in the form of 0-i=-i.

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Most popular questions from this chapter

Solve for all possible values of the real numbers x and y in the following equations.|x+iy|=y-ix. .

Find each of the following in rectangular a+zb form if z=2-z3; if z=x+zy.

1z-1

Solve for all possible values of the real numbersand in the following equations (x-iy)2=2ix.

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.

8.2e5πi/6

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,θ 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.

-1.

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