What is the general solution for sinh(x)= 6/5 ?

The general solution for sinh(x)= 6/5 is x=ln((6+sqrt(61))/5)

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sinh(x)= 6/5

Solution

sinh (x) = \frac{6}{5}

Solution

x = ln (\frac{6 + 61}{5})

Degrees

x = 58.21097 \dots^{\circ}

Solution steps

sinh (x) = \frac{6}{5}

Rewrite using trig identities

sinh (x) = \frac{6}{5}

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{6}{5}

\frac{e ^{x} - e ^{- x}}{2} = \frac{6}{5}

\frac{e ^{x} - e ^{- x}}{2} = \frac{6}{5} : x = ln (\frac{6 + 61}{5})

\frac{e ^{x} - e ^{- x}}{2} = \frac{6}{5}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 5 = 2 \cdot 6

Simplify

(e^{x} - e^{- x}) \cdot 5 = 12

Apply exponent rules

(e^{x} - e^{- x}) \cdot 5 = 12

Apply exponent rule:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 5 = 12

(e^{x} - (e^{x})^{- 1}) \cdot 5 = 12

Rewrite the equation with

e^{x} = u

(u - (u)^{- 1}) \cdot 5 = 12

Solve

(u - u^{- 1}) \cdot 5 = 12 : u = \frac{6 + 61}{5}, u = \frac{6 - 61}{5}

(u - u^{- 1}) \cdot 5 = 12

Refine

(u - \frac{1}{u}) \cdot 5 = 12

Simplify

(u - \frac{1}{u}) \cdot 5 : 5 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 5

Apply the commutative law:

(u - \frac{1}{u}) \cdot 5 = 5 (u - \frac{1}{u})

5 (u - \frac{1}{u})

5 (u - \frac{1}{u}) = 12

Expand

5 (u - \frac{1}{u}) : 5 u - \frac{5}{u}

5 (u - \frac{1}{u})

Apply the distributive law:

a (b - c) = ab - a c

a = 5, b = u, c = \frac{1}{u}

= 5 u - 5 \cdot \frac{1}{u}

5 \cdot \frac{1}{u} = \frac{5}{u}

5 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 5}{u}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{u}

= 5 u - \frac{5}{u}

5 u - \frac{5}{u} = 12

Multiply both sides by

u

5 u - \frac{5}{u} = 12

Multiply both sides by

u

5 uu - \frac{5}{u} u = 12 u

Simplify

5 uu - \frac{5}{u} u = 12 u

Simplify

5 uu : 5 u^{2}

5 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 5 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 5 u^{2}

Simplify

- \frac{5}{u} u : - 5

- \frac{5}{u} u

Multiply fractions:

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

= - \frac{5 u}{u}

Cancel the common factor:

u

= - 5

5 u^{2} - 5 = 12 u

5 u^{2} - 5 = 12 u

5 u^{2} - 5 = 12 u

Solve

5 u^{2} - 5 = 12 u : u = \frac{6 + 61}{5}, u = \frac{6 - 61}{5}

5 u^{2} - 5 = 12 u

Move

12 u

to the left side

5 u^{2} - 5 = 12 u

Subtract

12 u

from both sides

5 u^{2} - 5 - 12 u = 12 u - 12 u

Simplify

5 u^{2} - 5 - 12 u = 0

5 u^{2} - 5 - 12 u = 0

Write in the standard form

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

5 u^{2} - 12 u - 5 = 0

Solve with the quadratic formula

5 u^{2} - 12 u - 5 = 0

Quadratic Equation Formula:

For

a = 5, b = - 12, c = - 5

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 5 ( - 5 )}{2 \cdot 5}

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 5 ( - 5 )}{2 \cdot 5}

(- 12)^{2} - 4 \cdot 5 (- 5) = 261

(- 12)^{2} - 4 \cdot 5 (- 5)

Apply rule

- (- a) = a

= (- 12)^{2} + 4 \cdot 5 \cdot 5

Apply exponent rule:

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

n

is even

(- 12)^{2} = 1 2^{2}

= 1 2^{2} + 4 \cdot 5 \cdot 5

Multiply the numbers:

4 \cdot 5 \cdot 5 = 100

= 1 2^{2} + 100

1 2^{2} = 144

= 144 + 100

Add the numbers:

144 + 100 = 244

= 244

Prime factorization of

244 : 2^{2} \cdot 61

244

244

divides by

2244 = 122 \cdot 2

= 2 \cdot 122

122

divides by

2122 = 61 \cdot 2

= 2 \cdot 2 \cdot 61

2, 61

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 61

= 2^{2} \cdot 61

= 2^{2} \cdot 61

Apply radical rule:

= 61 2^{2}

Apply radical rule:

2^{2} = 2

= 261

u_{1, 2} = \frac{- ( - 12 ) \pm 2 61}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- ( - 12 ) + 2 61}{2 \cdot 5}, u_{2} = \frac{- ( - 12 ) - 2 61}{2 \cdot 5}

u = \frac{- ( - 12 ) + 2 61}{2 \cdot 5} : \frac{6 + 61}{5}

\frac{- ( - 12 ) + 2 61}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{12 + 2 61}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{12 + 2 61}{10}

Factor

12 + 261 : 2 (6 + 61)

12 + 261

Rewrite as

= 2 \cdot 6 + 261

Factor out common term

2

= 2 (6 + 61)

= \frac{2 ( 6 + 61 )}{10}

Cancel the common factor:

2

= \frac{6 + 61}{5}

u = \frac{- ( - 12 ) - 2 61}{2 \cdot 5} : \frac{6 - 61}{5}

\frac{- ( - 12 ) - 2 61}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{12 - 2 61}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{12 - 2 61}{10}

Factor

12 - 261 : 2 (6 - 61)

12 - 261

Rewrite as

= 2 \cdot 6 - 261

Factor out common term

2

= 2 (6 - 61)

= \frac{2 ( 6 - 61 )}{10}

Cancel the common factor:

2

= \frac{6 - 61}{5}

The solutions to the quadratic equation are:

u = \frac{6 + 61}{5}, u = \frac{6 - 61}{5}

u = \frac{6 + 61}{5}, u = \frac{6 - 61}{5}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u - u^{- 1}) 5

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{6 + 61}{5}, u = \frac{6 - 61}{5}

u = \frac{6 + 61}{5}, u = \frac{6 - 61}{5}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{6 + 61}{5} : x = ln (\frac{6 + 61}{5})

e^{x} = \frac{6 + 61}{5}

Apply exponent rules

e^{x} = \frac{6 + 61}{5}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{6 + 61}{5})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{6 + 61}{5})

x = ln (\frac{6 + 61}{5})

Solve

e^{x} = \frac{6 - 61}{5} :

No Solution for

x \in R

e^{x} = \frac{6 - 61}{5}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (\frac{6 + 61}{5})

x = ln (\frac{6 + 61}{5})

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)= 6/5 ?
The general solution for sinh(x)= 6/5 is x=ln((6+sqrt(61))/5)