What is the general solution for sinh(x)=4 ?

The general solution for sinh(x)=4 is x=ln(4+sqrt(17))

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sinh(x)=4

Solution

sinh (x) = 4

Solution

x = ln (4 + 17)

Degrees

x = 120.01818 \dots^{\circ}

Solution steps

sinh (x) = 4

Rewrite using trig identities

sinh (x) = 4

Use the Hyperbolic identity:

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

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

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

\frac{e ^{x} - e ^{- x}}{2} = 4 : x = ln (4 + 17)

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

Multiply both sides by

2

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

Simplify

e^{x} - e^{- x} = 8

Apply exponent rules

e^{x} - e^{- x} = 8

Apply exponent rule:

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

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

e^{x} - (e^{x})^{- 1} = 8

e^{x} - (e^{x})^{- 1} = 8

Rewrite the equation with

e^{x} = u

u - (u)^{- 1} = 8

Solve

u - u^{- 1} = 8 : u = 4 + 17, u = 4 - 17

u - u^{- 1} = 8

Refine

u - \frac{1}{u} = 8

Multiply both sides by

u

u - \frac{1}{u} = 8

Multiply both sides by

u

uu - \frac{1}{u} u = 8 u

Simplify

uu - \frac{1}{u} u = 8 u

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

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

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

Cancel the common factor:

u

= - 1

u^{2} - 1 = 8 u

u^{2} - 1 = 8 u

u^{2} - 1 = 8 u

Solve

u^{2} - 1 = 8 u : u = 4 + 17, u = 4 - 17

u^{2} - 1 = 8 u

Move

8 u

to the left side

u^{2} - 1 = 8 u

Subtract

8 u

from both sides

u^{2} - 1 - 8 u = 8 u - 8 u

Simplify

u^{2} - 1 - 8 u = 0

u^{2} - 1 - 8 u = 0

Write in the standard form

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

u^{2} - 8 u - 1 = 0

Solve with the quadratic formula

u^{2} - 8 u - 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 8, c = - 1

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

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

(- 8)^{2} - 4 \cdot 1 \cdot (- 1) = 217

(- 8)^{2} - 4 \cdot 1 \cdot (- 1)

Apply rule

- (- a) = a

= (- 8)^{2} + 4 \cdot 1 \cdot 1

Apply exponent rule:

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

n

is even

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

= 8^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 8^{2} + 4

8^{2} = 64

= 64 + 4

Add the numbers:

64 + 4 = 68

= 68

Prime factorization of

68 : 2^{2} \cdot 17

68

68

divides by

268 = 34 \cdot 2

= 2 \cdot 34

34

divides by

234 = 17 \cdot 2

= 2 \cdot 2 \cdot 17

2, 17

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 17

= 2^{2} \cdot 17

= 2^{2} \cdot 17

Apply radical rule:

= 17 2^{2}

Apply radical rule:

2^{2} = 2

= 217

u_{1, 2} = \frac{- ( - 8 ) \pm 2 17}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 2 17}{2 \cdot 1}, u_{2} = \frac{- ( - 8 ) - 2 17}{2 \cdot 1}

u = \frac{- ( - 8 ) + 2 17}{2 \cdot 1} : 4 + 17

\frac{- ( - 8 ) + 2 17}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{8 + 2 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{8 + 2 17}{2}

Factor

8 + 217 : 2 (4 + 17)

8 + 217

Rewrite as

= 2 \cdot 4 + 217

Factor out common term

2

= 2 (4 + 17)

= \frac{2 ( 4 + 17 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 4 + 17

u = \frac{- ( - 8 ) - 2 17}{2 \cdot 1} : 4 - 17

\frac{- ( - 8 ) - 2 17}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{8 - 2 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{8 - 2 17}{2}

Factor

8 - 217 : 2 (4 - 17)

8 - 217

Rewrite as

= 2 \cdot 4 - 217

Factor out common term

2

= 2 (4 - 17)

= \frac{2 ( 4 - 17 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 4 - 17

The solutions to the quadratic equation are:

u = 4 + 17, u = 4 - 17

u = 4 + 17, u = 4 - 17

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u - u^{- 1}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 4 + 17, u = 4 - 17

u = 4 + 17, u = 4 - 17

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 4 + 17 : x = ln (4 + 17)

e^{x} = 4 + 17

Apply exponent rules

e^{x} = 4 + 17

f (x) = g (x)

, then

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

ln (e^{x}) = ln (4 + 17)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (4 + 17)

x = ln (4 + 17)

Solve

e^{x} = 4 - 17 :

No Solution for

x \in R

e^{x} = 4 - 17

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (4 + 17)

x = ln (4 + 17)

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)=4 ?
The general solution for sinh(x)=4 is x=ln(4+sqrt(17))