Q19P Verify the formulas in Problems ... [FREE SOLUTION]

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Chapter 2: Q19P (page 81)

Verify the formulas in Problems 17 to 24.
$a r c t a n z = \frac{1}{2 i} \frac{l n 1 + i z}{1 - i z}$

Short Answer

Expert verified

The formula is verified $a r c t a n z = \frac{1}{2 i} \frac{\ln 1 + iz}{1 - iz}$ .

Step by step solution

Given Information

Given formula is $a r c t a n z = \frac{1}{2 i} \frac{\ln 1 + iz}{1 - iz}$

Definition of Trigonometric equation.

A trigonometric equation is one that has one or more trigonometric ratios with unknown angles.

Use exponential form to expand the equation

Given the function $a r c t a n z = \frac{1}{2 i} \frac{\ln 1 + iz}{1 - iz}$ .

Write the exponential form of the $\tan (w) = z .$ .

$\tan (w) = \frac{\sin (w)}{\cos (w)} = \frac{e x p (w i) - e x p (- w i)}{i [e x p (w i) + e x p (- w i)]}$

role="math" localid="1658900771717" $= \frac{e x p (w i) - e x p (- w i)}{i [e x p (w i) + e x p (- w i)]} = z . . . . . (1)$

Rewrite the equation (1).

role="math" localid="1658900849442" $e^{(w i)} - e^{(- w i)} = z [e^{(w i)} + e^{(- w i)}] e^{(2 w i)} - 1 = i z [e^{(2 w i)} + 1] e^{(2 w i)} - 1 = i z e^{(2 w i)} - 1 + i z e^{(2 w i)} [1 - i z] = [1 + i z] e^{(2 w i)} = \frac{1 + i z}{1 - i z} . . . . . . . (2)$

Simplify equation (2).

$2 w i = \ln (\frac{1 + i z}{1 - i z}) w = \frac{\ln (\frac{1 + i z}{1 - i z})}{2 i} w = \frac{1}{2 i} \ln \frac{1 + i z}{1 - i z}$

Replace value of by $w = a r c \tan (z)$ .

role="math" localid="1658901451189" $a r c t a n z = \frac{1}{2 i} \ln \frac{1 + i z}{1 - i z}$

Hence the formula is verified

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91影视

Verify the formulas in Problems 17 to 24.
$a r c t a n z = \frac{1}{2 i} \frac{l n 1 + i z}{1 - i z}$

Short Answer

Step by step solution

Given Information

Definition of Trigonometric equation.

Use exponential form to expand the equation

One App. One Place for Learning.

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91影视

Verify the formulas in Problems 17 to 24.arctanz=12iln1+iz1-iz

Short Answer

Step by step solution

Given Information

Definition of Trigonometric equation.

Use exponential form to expand the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Classical Mechanics

Atoms and Radioactivity

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Engineering Physics

Study anywhere. Anytime. Across all devices.

Verify the formulas in Problems 17 to 24.
$a r c t a n z = \frac{1}{2 i} \frac{l n 1 + i z}{1 - i z}$