chapter 1: hyperbolic, inverse hyperbolic, inverse trigonometric

7/28/2019 CHAPTER 1: HYPERBOLIC, INVERSE HYPERBOLIC, INVERSE TRIGONOMETRIC

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1.1: COMPREHEND HYPERBOLIC FUNCTIONS

1.1.1: Define Hyperbolic Functions

Combinations of exponential functions (ex) Form of: sinh, cosh, tanh, cosech, sech and coth

1.1.2: Calculate Values of Hyperbolic Functions

Example (a)Find the value of:

i. sinh (1.275)ii. cosh (2.156)

iii. tanh (ln 3)Solution:

i. sinh (1.275)Step 1: Identify the formula

2 xx eexsinh

Step 2: Replace the value ofx into the formula

x = 1.275 2

1.275275.1275.1

ee

sinh = 1.65

HYPERBOLIC, INVERSE HYPERBOLIC &

INVERSE TRIGONOMETRIC FUNCTIONS

FORMULA:


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ii. cosh (2.156)2

xxee

xcosh

2156.2cosh

156.2156.2

ee= 4.376

iii. tanh (ln 3)

xx

xx

ee

eextanh

3ln3ln

3ln3ln

)3tanh(ln

ee

ee= 0.8

1.1.3: Sketch Graphs of Hyperbolic Functions

HYPERBOLIC FUNCTIONS GRAPH

sinh

cosh

tanh

Tr y this!

Find to 4 decimal places for:

a) cosech (-2.5)

b) sech (-ln 2)

c) coth (0.38)

d)

HYPERBOLIC FUNCTIONS GRAPH

cosech

sech

coth

HYPERBOLIC IDENTITIES:

cosech = 1n

sech

=1

coth = 1n =n


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Example (b)Sketch the graph of y = coshx from point x = -5 to x = 5

Solution:

Step 1: Create table from start point to end point (using calculator)

x -5 -3 0 3 5

y 74 10 1 10 74

Step 2: Sketch graph by referring to the table

1.1.4: Prove Identities of Hyperbolic Functions

Example (c)Prove that:

i. 2 sinh (x +y) cosh (xy) = sinh 2x + sinh 2yii. coshxsinhx = e-x

iii. 1tanh2x = sech2x

FORMULA:

74

10

3 5-5 -3

Try this!

a) A curve line is based on . If c = 50, Calculate the value of y when x = 109.

b) Fill in the table below using . Then sketch a graph.

x 2.5 2.9 3.3 3.7 4.1 4.5

y


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Solution:

i. 2 sinh (x +y) cosh (xy) = sinh 2x + sinh 2y

Step 1: Expand the equation based on formula.

2 sinh (x +y) cosh (xy) =

Step 2: Simplify the calculations until proven.

2 sinh (x +y) cosh (xy) =

=

=

= sinh 2x + sinh 2y

ii. coshxsinhx= e-x

=

=

=

= e-x

iii. 1tanh2x = sech2x=

=

=

=

=

= sech2x

Try this!

Verify that:

a) sinh2xsinh2y = cosh2xcosh2y

b) cosh 2x = 1 + 2 sinh2x


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1.2: KNOW INVERSE HYPERBOLIC FUNCTIONS

1.2.1: Define Inverse Hyperbolic Function

The inverses of the hyperbolic functions are the area hyperbolic functions. Form of sinh-1, cosh-1, tanh-1, cosech-1, sech-1 and coth-1 The inverse hyperbolic functions are given by:

i. y = tanh-1x x = tanhyii. y = sinh

-1

x x = sinhyiii. y = cosh-1x x = coshyiv. y = coth-1x x = cothyv. y = cosech-1x x = cosechy

vi. y = sech-1x x = sechy

1.2.2: CalculateValues of Inverse Hyperbolic Functions

Example (d)Find to 4 decimal places:

i. cosh-1 (2.364)ii. tanh-1 (-0.322)

iii. sinh-1x = ln 2

Solution:

i. cosh-1 (2.364)Step 1: Identify the formula

1ln 21 xxxcosh

Step 2: Replace the value ofx into the formula

x = 2.364

1364.2364.2ln364.2 21cosh

= = 1.5054

FORMULA:


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ii. tanh-1 (-0.322)

x

xxtanh

1

1ln

2

11

)322.0(1

)322.0(1ln

2

1322.0

1tanh

= 1 = -0.3339

iii. sinh-1x = ln 2 =

=

( ) =

x2 + 1 = 44x +x2

x = 0.75

1.2.3: Sketch The Graphs of Inverse Hyperbolic Functions

INVERSE HYPERBOLIC

FUNCTIONSGRAPH

sinh-1

cosh-1

tanh-1

Try this!

Evaluate:

a) cosech-13b) coth-1 (5)

INVERSE HYPERBOLIC

FUNCTIONSGRAPH

cosech-1

sech-1

coth-1

INVERSE HYPERBOLIC

IDENTITIES:

cosech1 =sih1 1

sech1 =cosh1 1

coth1 =tah1 1


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Example (e)Sketch the graph of y = sinh-1x from point x = -30 to x = 30

Solution:

Step 1: Create table from start point to end point (using calculator)

x -30 -10 0 10 30

y -4 -3 0 3 4

Step 2: Sketch graph by referring to the table

1.2.4: Derive The Formulae of Inverse Hyperbolic Functions

Since the hyperbola functions are in essence exponential functions you should expect their inverses to involve the natural logarithm. Example (f)

Prove that sinh-1x = ln( )

Solution:

Step 1: Switch the equationy = sinh-1x x = sinhy

Step 2: Choose formula from page 1 (unit 1.1) and simplify

2sinh

yyee

yx

= yy ee

-3

-4

4

3

10 30-30 -10

Try this!

Sketch graphs for below functions:

a) y = sech-1x from point x = 0.1 to x = 1

b) cosh

-1


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Step 3: Multiply with ey

= yy ee = =

Step 4: Use the Quadratic Formula to solve this quadratic-like equation.

=

a =1 , b = -2x , c = -1 , x = ey

=

= =

Step 5: Simplify using logarithm law

=

=

= ( ) sinh-1x = ln( ) = Proved!

1.3: RECOGNIZE INVERSE TRIGONOMETRIC FUNCTIONS

1.3.1: Define Inverse Trigonometric Functions

The inverse functions of the trigonometric functions Form of cos-1, cot-1, cosec-1, sec-1, sin-1, and tan-1 The inverse trigonometric functions are given by:

i. y = tan-1xx = tanyii. y = sin-1xx = siny

iii. y = cos-1xx = cosyiv. y = cot-1xx = cotyv. y = cosec-1xx = cosecy

vi. y = sec-1xx = secy

Try this!

Verify that:

a)

tah1

=

1

1

1for

<

b)


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1.3.2: Find The Principal Values of Inverse Trigonometric Functions

Example (g)Find the principal values of:

i. tan-1 13ii. cos-1 (-0.1587)iii. cosec-1 (1.11)

Solution:

i. tan-1 13Step 1: Find the value using calculator (in degreeo and radian)

tan-1 13= 30oand 0.524 rad

ii. cos-1 (-0.1587) = 99.13oand 1.73 radiii. cosec-1 (1.11) = sin-1 1111 = 64.277oand 1.122 rad

1.3.3: Solve Equations Involving Inverse Trigonometric Functions

Example (h)Simplify:

i. tasi1 ii. costa1

Solution:

i. tasi1 Step 1: Let inverse trigonometric (x) =y

si1 = y

Step 2: Then x = trigonometricy (remember that when youre putting sin-1 on the other side of the equality, you remove the -1).

= sin y

Step 3: Draw a triangle ABC and complete the value using the Pythagorean theorem

Try this!

Determine the principal value of the following functions:

a) sin-13

b) tan-1 (-0.45)

x5

A B

C

AC + AB = BCx2 - 52 = -AB2

AB2 = 52 - x2

AB =


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1

Step 4: Complete the equation.

tasi1 = tany=

ii. ta1 = y= tan y

costa1 = cos y=

5

x

Try this!

Simplify:

a) cos2 si1

b) sin 2sec1 3

chapter 1: hyperbolic, inverse hyperbolic, inverse trigonometric

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