Chapter 2: Q9P (page 76)

Find each of the following in the $x + i y$ form and compare a computer solution.
$t a n h^{-} (i \sqrt{3})$ .

Short Answer

Expert verified

The $x + i y$ form of the given equation $\tan h^{- 1} (i \sqrt{3})$ is $i (\frac{π}{3} \pm n π)$

Step by step solution

Given Information.

The given expression is, $\tan h^{- 1} (i \sqrt{3})$ .

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 is, $z = \tan h^{- 1} (i \sqrt[]{3})$ .

Rewrite the above expression.

$\tan h z = (i \sqrt{3})$

Write the formula for $(θ)$ .

$\frac{e^{(z)} - e^{(- z)}}{e^{(z)} + e^{(- z)}} = i \sqrt{3}$

Multiply it by $\frac{e^{(z)}}{e^{(z)}}$ .

$e^{(2 z)} - 1 = i {\sqrt{3 e}}^{2 z} + i \sqrt{3} e^{(2 z)} [1 - i \sqrt{3}] = [1 + i \sqrt{3}] e^{(2 z)} = \frac{1 + i \sqrt{3}}{1 i \sqrt{3}}$

Convert in rectangular form

Take the logarithm function both sides.

$2 z = I n \frac{1 + i \sqrt{3}}{1 - i \sqrt{3}} 2 z = I n (\frac{2 e^{(π / 3)}}{2 e^{(- π / 3)}}) 2 z = i \frac{2 π}{3} \pm 2 n π$

$2 z = i (\frac{2 π}{3} \pm 2 n π)$ Where $n = 0, 1, 2, 3, . .$

$z = i (\frac{π}{3} \pm n π)$ . Where $n = 0, 1, 2, 3, . .$ $n = 0, 1, 2, 3, . . .$

Hence the general solution is $z = i (\frac{π}{3} \pm n π)$ .

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Find each of the following in the $x + i y$ form and compare a computer solution.
$t a n h^{-} (i \sqrt{3})$ .

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Convert into polar form.

Convert in rectangular form

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Find each of the following in the x+iy form and compare a computer solution.tanh-(i3).

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

Kinematics Physics

Measurements

Modern Physics

Magnetism

Waves Physics

Fluids

Study anywhere. Anytime. Across all devices.

Find each of the following in the $x + i y$ form and compare a computer solution.
$t a n h^{-} (i \sqrt{3})$ .