What is the general solution for cosh(x)cosh(1)-sinh(x)sinh(1)=2 ?

The general solution for cosh(x)cosh(1)-sinh(x)sinh(1)=2 is x=ln(e(2+sqrt(3))),x=ln(e(2-sqrt(3)))

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cosh(x)cosh(1)-sinh(x)sinh(1)=2

Solution

cosh (x) cosh (1) - sinh (x) sinh (1) = 2

Solution

x = ln (e (2 + 3)), x = ln (e (2 - 3))

Degrees

x = 132.75190 \dots^{\circ}, x = - 18.16034 \dots^{\circ}

Solution steps

cosh (x) cosh (1) - sinh (x) sinh (1) = 2

Rewrite using trig identities

cosh (x) cosh (1) - sinh (x) sinh (1) = 2

Use the Hyperbolic identity:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

cosh (x) cosh (1) - \frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e ^{1} - e ^{- 1}}{2} = 2

Use the Hyperbolic identity:

cosh (x) = \frac{e ^{x} + e ^{- x}}{2}

\frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e ^{1} + e ^{- 1}}{2} - \frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e ^{1} - e ^{- 1}}{2} = 2

\frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e ^{1} + e ^{- 1}}{2} - \frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e ^{1} - e ^{- 1}}{2} = 2

\frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e ^{1} + e ^{- 1}}{2} - \frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e ^{1} - e ^{- 1}}{2} = 2 : x = ln (e (2 + 3)), x = ln (e (2 - 3))

\frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e ^{1} + e ^{- 1}}{2} - \frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e ^{1} - e ^{- 1}}{2} = 2

Multiply both sides by

4

\frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e ^{1} + e ^{- 1}}{2} \cdot 4 - \frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e ^{1} - e ^{- 1}}{2} \cdot 4 = 2 \cdot 4

Simplify

\frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e ^{1} + e ^{- 1}}{2} \cdot 4 - \frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e ^{1} - e ^{- 1}}{2} \cdot 4 : \frac{( e ^{2} + 1 ) ( e ^{x} + e ^{- x} )}{e} - \frac{( e ^{2} - 1 ) ( e ^{x} - e ^{- x} )}{e}

\frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e ^{1} + e ^{- 1}}{2} \cdot 4 - \frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e ^{1} - e ^{- 1}}{2} \cdot 4

Apply rule

a^{1} = a

e^{1} = e

= 4 \cdot \frac{e ^{- 1} + e}{2} \cdot \frac{e ^{x} + e ^{- x}}{2} - 4 \cdot \frac{- e ^{- 1} + e}{2} \cdot \frac{e ^{x} - e ^{- x}}{2}

\frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e + e ^{- 1}}{2} \cdot 4 = \frac{( e ^{2} + 1 ) ( e ^{x} + e ^{- x} )}{e}

\frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e + e ^{- 1}}{2} \cdot 4

\frac{e + e ^{- 1}}{2} = \frac{e ^{2} + 1}{2 e}

\frac{e + e ^{- 1}}{2}

Apply exponent rule:

a^{- 1} = \frac{1}{a}

= \frac{e + \frac{1}{e}}{2}

Join

e + \frac{1}{e} : \frac{e ^{2} + 1}{e}

e + \frac{1}{e}

Convert element to fraction:

e = \frac{ee}{e}

= \frac{ee}{e} + \frac{1}{e}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{ee + 1}{e}

ee + 1 = e^{2} + 1

ee + 1

ee = e^{2}

ee

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

ee = e^{1 + 1}

= e^{1 + 1}

Add the numbers:

1 + 1 = 2

= e^{2}

= e^{2} + 1

= \frac{e ^{2} + 1}{e}

= \frac{\frac{e ^{2} + 1}{e}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{e ^{2} + 1}{e 2}

= 4 \cdot \frac{e ^{2} + 1}{2 e} \cdot \frac{e ^{x} + e ^{- x}}{2}

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{( e ^{x} + e ^{- x} ) ( e ^{2} + 1 ) \cdot 4}{2 e 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 ( e ^{2} + 1 ) ( e ^{x} + e ^{- x} )}{4 e}

Divide the numbers:

\frac{4}{4} = 1

= \frac{( e ^{2} + 1 ) ( e ^{x} + e ^{- x} )}{e}

\frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e - e ^{- 1}}{2} \cdot 4 = \frac{( e ^{2} - 1 ) ( e ^{x} - e ^{- x} )}{e}

\frac{e ^{x} - e ^{- x}}{2} \cdot \frac{e - e ^{- 1}}{2} \cdot 4

\frac{e - e ^{- 1}}{2} = \frac{e ^{2} - 1}{2 e}

\frac{e - e ^{- 1}}{2}

Apply exponent rule:

a^{- 1} = \frac{1}{a}

= \frac{e - \frac{1}{e}}{2}

Join

e - \frac{1}{e} : \frac{e ^{2} - 1}{e}

e - \frac{1}{e}

Convert element to fraction:

e = \frac{ee}{e}

= \frac{ee}{e} - \frac{1}{e}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{ee - 1}{e}

ee - 1 = e^{2} - 1

ee - 1

ee = e^{2}

ee

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

ee = e^{1 + 1}

= e^{1 + 1}

Add the numbers:

1 + 1 = 2

= e^{2}

= e^{2} - 1

= \frac{e ^{2} - 1}{e}

= \frac{\frac{e ^{2} - 1}{e}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{e ^{2} - 1}{e 2}

= 4 \cdot \frac{e ^{2} - 1}{2 e} \cdot \frac{e ^{x} - e ^{- x}}{2}

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{( e ^{x} - e ^{- x} ) ( e ^{2} - 1 ) \cdot 4}{2 e 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 ( e ^{2} - 1 ) ( e ^{x} - e ^{- x} )}{4 e}

Divide the numbers:

\frac{4}{4} = 1

= \frac{( e ^{2} - 1 ) ( e ^{x} - e ^{- x} )}{e}

= \frac{( e ^{2} + 1 ) ( e ^{x} + e ^{- x} )}{e} - \frac{( e ^{2} - 1 ) ( e ^{x} - e ^{- x} )}{e}

\frac{( e ^{2} + 1 ) ( e ^{x} + e ^{- x} )}{e} - \frac{( e ^{2} - 1 ) ( e ^{x} - e ^{- x} )}{e} = 8

Apply exponent rules

\frac{( e ^{2} + 1 ) ( e ^{x} + e ^{- x} )}{e} - \frac{( e ^{2} - 1 ) ( e ^{x} - e ^{- x} )}{e} = 8

Apply exponent rule:

a^{b c} = (a^{b})^{c}

e^{- x} = (e^{x})^{- 1}

\frac{( e ^{2} + 1 ) ( e ^{x} + ( e ^{x} ) ^{- 1} )}{e} - \frac{( e ^{2} - 1 ) ( e ^{x} - ( e ^{x} ) ^{- 1} )}{e} = 8

\frac{( e ^{2} + 1 ) ( e ^{x} + ( e ^{x} ) ^{- 1} )}{e} - \frac{( e ^{2} - 1 ) ( e ^{x} - ( e ^{x} ) ^{- 1} )}{e} = 8

Rewrite the equation with

e^{x} = u

\frac{( e ^{2} + 1 ) ( u + ( u ) ^{- 1} )}{e} - \frac{( e ^{2} - 1 ) ( u - ( u ) ^{- 1} )}{e} = 8

Solve

\frac{( e ^{2} + 1 ) ( u + u ^{- 1} )}{e} - \frac{( e ^{2} - 1 ) ( u - u ^{- 1} )}{e} = 8 : u = e (2 + 3), u = e (2 - 3)

\frac{( e ^{2} + 1 ) ( u + u ^{- 1} )}{e} - \frac{( e ^{2} - 1 ) ( u - u ^{- 1} )}{e} = 8

Refine

\frac{( e ^{2} + 1 ) ( u ^{2} + 1 )}{e u} - \frac{( e ^{2} - 1 ) ( u ^{2} - 1 )}{e u} = 8

Multiply both sides by

e u

\frac{( e ^{2} + 1 ) ( u ^{2} + 1 )}{e u} - \frac{( e ^{2} - 1 ) ( u ^{2} - 1 )}{e u} = 8

Multiply both sides by

e u

\frac{( e ^{2} + 1 ) ( u ^{2} + 1 )}{e u} e u - \frac{( e ^{2} - 1 ) ( u ^{2} - 1 )}{e u} e u = 8 e u

Simplify

\frac{( e ^{2} + 1 ) ( u ^{2} + 1 )}{e u} e u - \frac{( e ^{2} - 1 ) ( u ^{2} - 1 )}{e u} e u = 8 e u

Simplify

\frac{( e ^{2} + 1 ) ( u ^{2} + 1 )}{e u} e u : (e^{2} + 1) (u^{2} + 1)

\frac{( e ^{2} + 1 ) ( u ^{2} + 1 )}{e u} e u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( e ^{2} + 1 ) ( u ^{2} + 1 ) e u}{e u}

Cancel the common factor:

e

= \frac{( e ^{2} + 1 ) ( u ^{2} + 1 ) u}{u}

Cancel the common factor:

u

= (e^{2} + 1) (u^{2} + 1)

Simplify

- \frac{( e ^{2} - 1 ) ( u ^{2} - 1 )}{e u} e u : - (e^{2} - 1) (u^{2} - 1)

- \frac{( e ^{2} - 1 ) ( u ^{2} - 1 )}{e u} e u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{( e ^{2} - 1 ) ( u ^{2} - 1 ) e u}{e u}

Cancel the common factor:

e

= - \frac{( e ^{2} - 1 ) ( u ^{2} - 1 ) u}{u}

Cancel the common factor:

u

= - (e^{2} - 1) (u^{2} - 1)

(e^{2} + 1) (u^{2} + 1) - (e^{2} - 1) (u^{2} - 1) = 8 e u

(e^{2} + 1) (u^{2} + 1) - (e^{2} - 1) (u^{2} - 1) = 8 e u

(e^{2} + 1) (u^{2} + 1) - (e^{2} - 1) (u^{2} - 1) = 8 e u

Solve

(e^{2} + 1) (u^{2} + 1) - (e^{2} - 1) (u^{2} - 1) = 8 e u : u = e (2 + 3), u = e (2 - 3)

(e^{2} + 1) (u^{2} + 1) - (e^{2} - 1) (u^{2} - 1) = 8 e u

Expand

(e^{2} + 1) (u^{2} + 1) - (e^{2} - 1) (u^{2} - 1) : 2 u^{2} + 2 e^{2}

(e^{2} + 1) (u^{2} + 1) - (e^{2} - 1) (u^{2} - 1)

Expand

(e^{2} + 1) (u^{2} + 1) : e^{2} u^{2} + e^{2} + u^{2} + 1

(e^{2} + 1) (u^{2} + 1)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = e^{2}, b = 1, c = u^{2}, d = 1

= e^{2} u^{2} + e^{2} \cdot 1 + 1 \cdot u^{2} + 1 \cdot 1

= e^{2} u^{2} + 1 \cdot e^{2} + 1 \cdot u^{2} + 1 \cdot 1

Simplify

e^{2} u^{2} + 1 \cdot e^{2} + 1 \cdot u^{2} + 1 \cdot 1 : e^{2} u^{2} + e^{2} + u^{2} + 1

e^{2} u^{2} + 1 \cdot e^{2} + 1 \cdot u^{2} + 1 \cdot 1

Multiply:

1 \cdot e^{2} = e^{2}

= e^{2} u^{2} + e^{2} + 1 \cdot u^{2} + 1 \cdot 1

Multiply:

1 \cdot u^{2} = u^{2}

= e^{2} u^{2} + e^{2} + u^{2} + 1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= e^{2} u^{2} + e^{2} + u^{2} + 1

= e^{2} u^{2} + e^{2} + u^{2} + 1

= e^{2} u^{2} + e^{2} + u^{2} + 1 - (e^{2} - 1) (u^{2} - 1)

Expand

- (e^{2} - 1) (u^{2} - 1) : - e^{2} u^{2} + e^{2} + u^{2} - 1

Expand

(e^{2} - 1) (u^{2} - 1) : e^{2} u^{2} - e^{2} - u^{2} + 1

(e^{2} - 1) (u^{2} - 1)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = e^{2}, b = - 1, c = u^{2}, d = - 1

= e^{2} u^{2} + e^{2} (- 1) + (- 1) u^{2} + (- 1) (- 1)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= e^{2} u^{2} - 1 \cdot e^{2} - 1 \cdot u^{2} + 1 \cdot 1

Simplify

e^{2} u^{2} - 1 \cdot e^{2} - 1 \cdot u^{2} + 1 \cdot 1 : e^{2} u^{2} - e^{2} - u^{2} + 1

e^{2} u^{2} - 1 \cdot e^{2} - 1 \cdot u^{2} + 1 \cdot 1

Multiply:

1 \cdot e^{2} = e^{2}

= e^{2} u^{2} - e^{2} - 1 \cdot u^{2} + 1 \cdot 1

Multiply:

1 \cdot u^{2} = u^{2}

= e^{2} u^{2} - e^{2} - u^{2} + 1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= e^{2} u^{2} - e^{2} - u^{2} + 1

= e^{2} u^{2} - e^{2} - u^{2} + 1

= - (e^{2} u^{2} - e^{2} - u^{2} + 1)

Distribute parentheses

= - (e^{2} u^{2}) - (- e^{2}) - (- u^{2}) - (1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - e^{2} u^{2} + e^{2} + u^{2} - 1

= e^{2} u^{2} + e^{2} + u^{2} + 1 - e^{2} u^{2} + e^{2} + u^{2} - 1

Simplify

e^{2} u^{2} + e^{2} + u^{2} + 1 - e^{2} u^{2} + e^{2} + u^{2} - 1 : 2 u^{2} + 2 e^{2}

e^{2} u^{2} + e^{2} + u^{2} + 1 - e^{2} u^{2} + e^{2} + u^{2} - 1

Group like terms

= e^{2} u^{2} - e^{2} u^{2} + u^{2} + u^{2} + e^{2} + e^{2} + 1 - 1

Add similar elements:

e^{2} + e^{2} = 2 e^{2}

= e^{2} u^{2} - e^{2} u^{2} + u^{2} + u^{2} + 2 e^{2} + 1 - 1

Add similar elements:

e^{2} u^{2} - e^{2} u^{2} = 0

= u^{2} + u^{2} + 2 e^{2} + 1 - 1

Add similar elements:

u^{2} + u^{2} = 2 u^{2}

= 2 u^{2} + 2 e^{2} + 1 - 1

1 - 1 = 0

= 2 u^{2} + 2 e^{2}

= 2 u^{2} + 2 e^{2}

2 u^{2} + 2 e^{2} = 8 e u

Move

8 e u

to the left side

2 u^{2} + 2 e^{2} = 8 e u

Subtract

8 e u

from both sides

2 u^{2} + 2 e^{2} - 8 e u = 8 e u - 8 e u

Simplify

2 u^{2} + 2 e^{2} - 8 e u = 0

2 u^{2} + 2 e^{2} - 8 e u = 0

Write in the standard form

a x^{2} + b x + c = 0

2 u^{2} - 8 e u + 2 e^{2} = 0

Solve with the quadratic formula

2 u^{2} - 8 e u + 2 e^{2} = 0

Quadratic Equation Formula:

For

a = 2, b = - 8 e, c = 2 e^{2}

u_{1, 2} = \frac{- ( - 8 e ) \pm ( - 8 e ) ^{2} - 4 \cdot 2 \cdot 2 e ^{2}}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 8 e ) \pm ( - 8 e ) ^{2} - 4 \cdot 2 \cdot 2 e ^{2}}{2 \cdot 2}

(- 8 e)^{2} - 4 \cdot 2 \cdot 2 e^{2} = 43 e

(- 8 e)^{2} - 4 \cdot 2 \cdot 2 e^{2}

(- 8 e)^{2} = 8^{2} e^{2}

(- 8 e)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 8 e)^{2} = (8 e)^{2}

= (8 e)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 8^{2} e^{2}

4 \cdot 2 \cdot 2 e^{2} = 16 e^{2}

4 \cdot 2 \cdot 2 e^{2}

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 16 e^{2}

= 8^{2} e^{2} - 16 e^{2}

8^{2} = 64

= 64 e^{2} - 16 e^{2}

Add similar elements:

64 e^{2} - 16 e^{2} = 48 e^{2}

= 48 e^{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 48 e^{2}

48 = 43

48

Prime factorization of

48 : 2^{4} \cdot 3

48

48

divides by

248 = 24 \cdot 2

= 2 \cdot 24

24

divides by

224 = 12 \cdot 2

= 2 \cdot 2 \cdot 12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3

= 2^{4} \cdot 3

= 2^{4} \cdot 3

Apply radical rule:

= 3 2^{4}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 3

Refine

= 43

= 43 e^{2}

Apply radical rule:

assuming

a \geq 0

e^{2} = e

= 43 e

u_{1, 2} = \frac{- ( - 8 e ) \pm 4 3 e}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 8 e ) + 4 3 e}{2 \cdot 2}, u_{2} = \frac{- ( - 8 e ) - 4 3 e}{2 \cdot 2}

u = \frac{- ( - 8 e ) + 4 3 e}{2 \cdot 2} : e (2 + 3)

\frac{- ( - 8 e ) + 4 3 e}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{8 e + 4 3 e}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{8 e + 4 3 e}{4}

Factor

8 e + 43 e : 4 e (2 + 3)

8 e + 43 e

Rewrite as

= 2 \cdot 4 e + 4 e 3

Factor out common term

4 e

= 4 e (2 + 3)

= \frac{4 e ( 2 + 3 )}{4}

Divide the numbers:

\frac{4}{4} = 1

= e (2 + 3)

u = \frac{- ( - 8 e ) - 4 3 e}{2 \cdot 2} : e (2 - 3)

\frac{- ( - 8 e ) - 4 3 e}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{8 e - 4 3 e}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{8 e - 4 3 e}{4}

Factor

8 e - 43 e : 4 e (2 - 3)

8 e - 43 e

Rewrite as

= 2 \cdot 4 e - 4 e 3

Factor out common term

4 e

= 4 e (2 - 3)

= \frac{4 e ( 2 - 3 )}{4}

Divide the numbers:

\frac{4}{4} = 1

= e (2 - 3)

The solutions to the quadratic equation are:

u = e (2 + 3), u = e (2 - 3)

u = e (2 + 3), u = e (2 - 3)

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{( e ^{2} + 1 ) ( u + u ^{- 1} )}{e} - \frac{( e ^{2} - 1 ) ( u - u ^{- 1} )}{e}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = e (2 + 3), u = e (2 - 3)

u = e (2 + 3), u = e (2 - 3)

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = e (2 + 3) : x = ln (e (2 + 3))

e^{x} = e (2 + 3)

Apply exponent rules

e^{x} = e (2 + 3)

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (e (2 + 3))

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (e (2 + 3))

x = ln (e (2 + 3))

Solve

e^{x} = e (2 - 3) : x = ln (e (2 - 3))

e^{x} = e (2 - 3)

Apply exponent rules

e^{x} = e (2 - 3)

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (e (2 - 3))

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (e (2 - 3))

x = ln (e (2 - 3))

x = ln (e (2 + 3)), x = ln (e (2 - 3))

x = ln (e (2 + 3)), x = ln (e (2 - 3))

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2 cos (θ) = 2 cos (3 θ)

cos(t)+cos(2t)=-0.75

cos (t) + cos (2 t) = - 0.75

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Frequently Asked Questions (FAQ)

What is the general solution for cosh(x)cosh(1)-sinh(x)sinh(1)=2 ?
The general solution for cosh(x)cosh(1)-sinh(x)sinh(1)=2 is x=ln(e(2+sqrt(3))),x=ln(e(2-sqrt(3)))