Chapter 2: Q9P (page 71)

Verify each of the following by using equations (11.4), (12.2), and (12.3).
$\frac{d}{dz} \cos z = - \sin z$

Short Answer

Expert verified

The equation $\frac{d}{d z} \cos z = - \sin z$ is verified using the equations (11.4), (12.2) and (12.3).

Step by step solution

Given information

The given function is $\frac{d}{d z} \cos z = - \sin z$ .

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 exponential form to expand the equation

The exponential form of the given equation is,

$\cos z = \frac{e^{z i} + e^{- z i}}{2} (1)$ …. (1)

Differentiate equation (1) with respect to z.

$\frac{d}{d z} \cos (z) = \frac{1}{2} \times \frac{d}{d z} (e^{z i} + e^{- z i})$ …. (2)

Open derivative using algebra of derivatives.

$\frac{d}{d z} \cos (z) = \frac{1}{2} \times (e^{z i} - e^{- z i}) = \frac{1}{2} \times (e^{z i} - e^{- z i})$

Multiply the equation by $\frac{i}{i}$ .

$\frac{d}{d z} \cos (z) = \frac{i}{2} \times (e^{z i} - e^{- z i}) \times \frac{i}{i} \frac{d}{d z} \cos (z) = - \frac{(e^{z i} - e^{- z i})}{2 i} \frac{d}{d z} \cos (z) = - \sin z$

Hence the equation is verified.

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Verify each of the following by using equations (11.4), (12.2), and (12.3).
$\frac{d}{dz} \cos z = - \sin z$

Short Answer

Step by step solution

Given information

Definition of Hyperbolic Function.

Use exponential form 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).ddzcosz=-sinz

Short Answer

Step by step solution

Given information

Definition of Hyperbolic Function.

Use exponential form to expand the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

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Engineering Physics

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Electricity

Fields in Physics

Study anywhere. Anytime. Across all devices.

Verify each of the following by using equations (11.4), (12.2), and (12.3).
$\frac{d}{dz} \cos z = - \sin z$