Chapter 2: Q19P (page 71)

Verify each of the following by using equations (11.4), (12.2), and (12.3).
$t a n h z = \frac{t a n h x + i t a n y}{1 + i t a n h x t a n y}$

Short Answer

Expert verified

The equation $t a n h z = \frac{\tanh x + i \tan y}{1 + i \tanh x \tan y}$ is verified using the equations (11.4), (12.2) and (12.3).

Step by step solution

Given information

The given equation is, $t a n h z = \frac{\tanh x + i \tan y}{1 + i \tanh x \tan y}$ .

Definition of Hyperbolic Function.

A hyperbolic function is a representation of the relationship between a point's distances from the origin to the coordinate axes as a function of an angle.

Use the standard form of complex number to expand the equation

The given function is,

$t a n h z = \frac{\tanh x + i \tan y}{1 + i \tanh x \tan y}$ …. (1)

Put $z = x + i y$ in equation (1).

$\tan z = \tan (x - y i) \tan h (x + i y) = \frac{\sin h (x + y i)}{\cos h (x + y i)} = \frac{\sin (x) \cos (y) + i \cos h (x) \sin (y)}{\cos h (x) \cos (y) + i \sin h (x) \sin (y)}$

Divide numerator and denominator by $[\cos h (x) . \cos (y)]$ .

$\tan h (x + i y) = \frac{\sin (x) \cos (y) + i \cos h (x) \sin (y)}{\cos h (x) \cos (y) + i \sin h (x) \sin (y)} = \frac{[\sin h (x) \cos (y) + i \cos h (x) \sin (y)] / [\cos h (x) \cos (y)]}{[\cos h (x) \cos (y) + i \sin h (x) \sin (y)] / [\cos h (x) \cos (y)]}$

$\tan h (x + i y) = \frac{(\frac{\sin h (x) \cos (y)}{\cos (x) \cos (y)}) + (\frac{i \cos h (x) \sin (y)}{\cos h (x) \cos (y)})}{(\frac{\cos h (x) \cos (y)}{\cos h (x) \cos (y)}) + (\frac{i \sin h (x) \sin (y)}{\cos h (x) \cos (y)})} = \frac{\tan h (x) + i \tan (y)}{1 + i \tan h (x) \tan (y)}$

Hence the equation is verified.

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Verify each of the following by using equations (11.4), (12.2), and (12.3).
$t a n h z = \frac{t a n h x + i t a n y}{1 + i t a n h x t a n y}$

Short Answer

Step by step solution

Given information

Definition of Hyperbolic Function.

Use the standard form of complex number to expand the equation

One App. One Place for Learning.

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Verify each of the following by using equations (11.4), (12.2), and (12.3).tanhz=tanhx+itany1+itanhxtany

Short Answer

Step by step solution

Given information

Definition of Hyperbolic Function.

Use the standard form of complex number to expand the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electromagnetism

Conservation of Energy and Momentum

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Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Verify each of the following by using equations (11.4), (12.2), and (12.3).
$t a n h z = \frac{t a n h x + i t a n y}{1 + i t a n h x t a n y}$