Chapter 2: Q4P (page 76)

Question: Find each of the following in the x+iy form and compare a computer solution.
$\sinh^{- 1} (i / 2)$

Short Answer

Expert verified

The values of $z_{1}$ and $z_{2}$ are $i (\frac{π}{6} + 2 n π)$ and $i (\frac{5 π}{6} + 2 n π)$ respectively

Step by step solution

Given Information

Given expression is $\sinh^{- 1} (i / 2)$ .

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 is, $\sinh^{- 1} (i / 2)$ .

Write the exponential form of the sinh(x).

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

Let $u = e^{(z)}$ in equation (2).

$u - \frac{1}{u} = i u^{2} - i u - 1 = 0$

The coefficient of equation is as follows.

a = 1

b = -i

c = -1

Use quadratic formula to find roots.

$u = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} u_{1} = \frac{i + \sqrt{3}}{2} u_{2} = \frac{i - \sqrt{3}}{2}$

Find value of

Find $z_{1}$ by putting value of $u_{1}$

$z_{1} = In (u_{1}) = In (r) + i (θ + 2 n π) n = 0, 1, 2, 3, . . . . = In |\frac{i + \sqrt{3}}{2}| + i (θ + 2 n π) = In (1) + i (\frac{π}{6} + 2 n π) = i (\frac{π}{6} + 2 n π)$

Find $z_{2}$ by putting value of $u_{2}$

$z_{2} = In (u_{2}) = In (r) + i (θ + 2 n π) n = 0, 1, 2, 3, . . . . = In |\frac{i + \sqrt{3}}{2}| + i (θ + 2 n π) = In (1) + i (\frac{5 π}{6} + 2 n π) = i (\frac{5 π}{6} + 2 n π)$

Therefore, values of $z_{1}$ and $z_{2}$ are $i (\frac{π}{6} + 2 n π)$ and $i (\frac{5 π}{6} + 2 n π)$ respectively.

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Question: Find each of the following in the x+iy form and compare a computer solution.
$\sinh^{- 1} (i / 2)$

Short Answer

Step by step solution

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Definition of Trigonometric equation.

Use exponential form to expand the equation

Find value of

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Question: Find each of the following in the x+iy form and compare a computer solution.sinh-1(i/2)

Short Answer

Step by step solution

Given Information

Definition of Trigonometric equation.

Use exponential form to expand the equation

Find value of

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Work Energy and Power

Measurements

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Question: Find each of the following in the x+iy form and compare a computer solution.
$\sinh^{- 1} (i / 2)$