Chapter 2: Q20P (page 81)

Verify the formulas in Problems 17 to 24.
$s i n h^{- 1} z = l n (z \pm \sqrt{z^{2} + 1})$

Short Answer

Expert verified

The formula $\sinh^{- 1} z = \ln (z \pm \sqrt{z^{2} + 1})$ is verified.

Step by step solution

Given Information

Given formula is $\sinh^{- 1} z = \ln (z \pm \sqrt{z^{2} + 1})$ .

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 $\sinh^{- 1} z = \ln (z \pm \sqrt{z^{2} + 1})$ .

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

$z \frac{e^{(w)} - e^{(- w)}}{2} e^{(2 w)} - 2 e^{(w)} - 1 = 0$ …(1)

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

$u^{2} - 2 z u - 1 = 0$ …(2)

The coefficient of equation is as follows.

$a = 1 b = - 2 z c = - 1$

Use quadratic formula to find roots of equation (2).

role="math" localid="1658902359261" $u = \frac{- b \pm \sqrt{b^{2} - 4 a a c}}{2 a} = \frac{2 z \sqrt{{(- 2 z)}^{2} + 4}}{2} = z \pm \sqrt{z^{2} + 1}$

Replace value of u by $u = e^{(w)}$ .

$w = \ln (z \pm \sqrt{z^{2} + 1})$

Replace value of w by $w = s i n h^{- 1} (z) .$ .

$(z) = i \ln (z \pm \sqrt{z^{2} + 1})$

Hence the formula is verified.

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Verify the formulas in Problems 17 to 24.
$s i n h^{- 1} z = l n (z \pm \sqrt{z^{2} + 1})$

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

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Study anywhere. Anytime. Across all devices.

Verify the formulas in Problems 17 to 24.sinh-1z=ln(z±z2+1)

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

Quantum Physics

Force

Geometrical and Physical Optics

Linear Momentum

Wave Optics

Dynamics

Study anywhere. Anytime. Across all devices.

Verify the formulas in Problems 17 to 24.
$s i n h^{- 1} z = l n (z \pm \sqrt{z^{2} + 1})$