fdema001_ fdema001 - chapter 2

7/25/2019 Fdema001_ Fdema001 - Chapter 2

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HELP Matriculation CentreFoundation in Science FDEMA001Engineering Mathematics Applications MFH

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CHAPTER 2: HYPERBOLIC FUNCTIONS

2.1: INTRODUCTION

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric functions.

Derive hyperbolic functions from ordinary trigonometric functions.

In Chapter 1, a complex number could be written in either polar or exponential form, giving

rise to the equations:

If these two equations are added,

By replacing with,

where the right-hand side is entirely real numbers. Thus, this is called hyperbolic cosine:

Exponential function can be expressed as a series of powers of x,

If these the series subtract and divide by 2. This is called hyperbolic sine:

Both hyperbolic cosine and hyperbolic sine can also be represented using series of powers

of x:


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For hyperbolic tangent, the same calculation as ordinary trigonometric tangent applied:

2.2: GRAPHS OF HYPERBOLIC FUNCTIONS

2.3: EVALUATION OF HYPERBOLIC FUNCTIONS

The values of can be found using a calculator in just the same manneras the values of the circular trigonometric expressions were found.

The values can also be found using the exponential expressions.


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Example 2.1: Evaluate




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2.4: INVERSE HYPERBOLIC FUNCTIONS

The values of can be found using a calculator in just the samemanner as the values of the circular trigonometric expressions were found.

The values can also be found using the exponential expressions.





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2.5: LOG FORM OF THE INVERSE HYPERBOLIC FUNCTIONS

( )





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2.6: HYPERBOLIC IDENTITIES

Reciprocal hyperbolic functions

Other hyperbolic identities

--(1) --(2)(1) + (2) (1) (2) Multiply both expressions;

Divide by ;


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Divide by ;

Square both the expressions;

--(3) --(4)(3) (4)

(3) + (4)

Substitute

Substitute


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2.7: RELATIONSHIP BETWEEN TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS

From previous work on complex number,

--(5)

--(6)(5) + (6)

(5) (6)

From , substitute ,

From , substitute , Divide both sides by Note:


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From and

From and

Example 2.10: Prove that


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Example 2.11:Prove that


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TUTORIAL 2: HYPERBOLIC FUNCTIONS

1. On the same axes, draw sketch graphs of (a) (b) (c) .2. If find when and .3.

Calculate using exponential expressions, the value of:

a.

b.

4.

If , find and hence, evaluate x.5.

The curve assumed by a heavy chain or cable is given by . If calculate:a.

The value of when b. The value of when

6.

Simplify

.

7.

Prove that .8.

Express and in exponential form and hence solve for real values of x, theequation: .

9.

If and prove that .10.

Evaluate using logarithm:

a.

b.

11.

Prove that .12.Prove that .13.Prove that .14.Solve for real values of x: .15.

If calculate when and .

16.Prove that .

fdema001_ fdema001 - chapter 2

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