Chapter 2: Q15P (page 76)

Find each of the following in the x + iyform and compare a computer solution.
arctan(2 + i )

Short Answer

Expert verified

The x + iy form of the given equation $\arctan (2 + i) is \frac{i \ln (2)}{4} + (\frac{3 π}{8} \pm n π)$ is

Step by step solution

Given Information.

The given expression is $\arctan (2 + i) .$

Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x + iy in which x is the real part and y is the imaginary part.

Convert into polar form.

Consider the complex number $z = \arctan (2 + i)$ .

Rewrite the above expression.

$tanhz = 2 + i$

Write the formula for $\tanh (θ)$ .

$- i \frac{e^{(z)} - e^{(z)}}{e^{(z)} + e^{(z)}} = 2 + i$

Simplify

$e^{(z i)} - e^{(- z i)} = (2 i - 1) [e^{(z i)} + e^{(- z i)}] e^{(2 z i)} - 1 = (2 i - 1) e^{(2 z i)} + (2 i - 1) e^{(2 z i)} = \frac{i}{1 - i}$

Convert in rectangular form.

Take the logarithm function both sides.

$2 z i = (\frac{i}{1 - i}) 2 z = \ln (\frac{e x p (πi / 2)}{\sqrt{2} e x p (- πi / 4)}) 2 z = \ln (\frac{e x p (3 πi / 4)}{\sqrt{2}}) 2 z = \ln (\frac{1}{\sqrt{2}}) + i (\frac{3 π}{4} \pm 2 n π) 2 z = - \frac{\ln (2)}{2} + i (\frac{3 π}{4} \pm 2 n π)$

where n = 0,1,2,3,....

$z = \frac{2 z i}{2 i} z = \frac{- \frac{\ln (2)}{2} + i (\frac{3 π}{4} \pm 2 n π)}{2 i} z = \frac{i \ln (2)}{4} + (\frac{3 π}{8} \pm 2 n π)$

Hence the general solution equation arctan ( 2 + i ) is $\frac{i \ln (2)}{4} + (\frac{3 π}{8} \pm 2 n π)$ .

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Find each of the following in the x + iyform and compare a computer solution.
arctan(2 + i )

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Convert into polar form.

Convert in rectangular form.

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Most popular questions from this chapter

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Find each of the following in the x + iyform and compare a computer solution.arctan(2 + i )

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Convert into polar form.

Convert in rectangular form.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Space Physics

Solid State Physics

Scientific Method Physics

Force

Magnetism

Nuclear Physics

Study anywhere. Anytime. Across all devices.

Find each of the following in the x + iyform and compare a computer solution.
arctan(2 + i )