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

The general solution for sinh(x)=-6 is x=ln(-6+sqrt(37))

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sinh(x)=-6

Solution

sinh (x) = - 6

Solution

x = ln (- 6 + 37)

Degrees

x = - 142.76846 \dots^{\circ}

Solution steps

sinh (x) = - 6

Rewrite using trig identities

sinh (x) = - 6

Use the Hyperbolic identity:

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

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

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

\frac{e ^{x} - e ^{- x}}{2} = - 6 : x = ln (- 6 + 37)

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

Multiply both sides by

2

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

Simplify

e^{x} - e^{- x} = - 12

Apply exponent rules

e^{x} - e^{- x} = - 12

Apply exponent rule:

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

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

e^{x} - (e^{x})^{- 1} = - 12

e^{x} - (e^{x})^{- 1} = - 12

Rewrite the equation with

e^{x} = u

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

Solve

u - u^{- 1} = - 12 : u = - 6 + 37, u = - 6 - 37

u - u^{- 1} = - 12

Refine

u - \frac{1}{u} = - 12

Multiply both sides by

u

u - \frac{1}{u} = - 12

Multiply both sides by

u

uu - \frac{1}{u} u = - 12 u

Simplify

uu - \frac{1}{u} u = - 12 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 = - 12 u

u^{2} - 1 = - 12 u

u^{2} - 1 = - 12 u

Solve

u^{2} - 1 = - 12 u : u = - 6 + 37, u = - 6 - 37

u^{2} - 1 = - 12 u

Move

12 u

to the left side

u^{2} - 1 = - 12 u

Add

12 u

to both sides

u^{2} - 1 + 12 u = - 12 u + 12 u

Simplify

u^{2} - 1 + 12 u = 0

u^{2} - 1 + 12 u = 0

Write in the standard form

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

u^{2} + 12 u - 1 = 0

Solve with the quadratic formula

u^{2} + 12 u - 1 = 0

Quadratic Equation Formula:

For

a = 1, b = 12, c = - 1

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

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

1 2^{2} - 4 \cdot 1 \cdot (- 1) = 237

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

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 1 2^{2} + 4

1 2^{2} = 144

= 144 + 4

Add the numbers:

144 + 4 = 148

= 148

Prime factorization of

148 : 2^{2} \cdot 37

148

148

divides by

2148 = 74 \cdot 2

= 2 \cdot 74

74

divides by

274 = 37 \cdot 2

= 2 \cdot 2 \cdot 37

2, 37

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 37

= 2^{2} \cdot 37

= 2^{2} \cdot 37

Apply radical rule:

= 37 2^{2}

Apply radical rule:

2^{2} = 2

= 237

u_{1, 2} = \frac{- 12 \pm 2 37}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 12 + 2 37}{2 \cdot 1}, u_{2} = \frac{- 12 - 2 37}{2 \cdot 1}

u = \frac{- 12 + 2 37}{2 \cdot 1} : - 6 + 37

\frac{- 12 + 2 37}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 12 + 2 37}{2}

Factor

- 12 + 237 : 2 (- 6 + 37)

- 12 + 237

Rewrite as

= - 2 \cdot 6 + 237

Factor out common term

2

= 2 (- 6 + 37)

= \frac{2 ( - 6 + 37 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 6 + 37

u = \frac{- 12 - 2 37}{2 \cdot 1} : - 6 - 37

\frac{- 12 - 2 37}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 12 - 2 37}{2}

Factor

- 12 - 237 : - 2 (6 + 37)

- 12 - 237

Rewrite as

= - 2 \cdot 6 - 237

Factor out common term

2

= - 2 (6 + 37)

= - \frac{2 ( 6 + 37 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - (6 + 37)

Negate

- (6 + 37) = - 6 - 37

= - 6 - 37

The solutions to the quadratic equation are:

u = - 6 + 37, u = - 6 - 37

u = - 6 + 37, u = - 6 - 37

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 = - 6 + 37, u = - 6 - 37

u = - 6 + 37, u = - 6 - 37

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = - 6 + 37 : x = ln (- 6 + 37)

e^{x} = - 6 + 37

Apply exponent rules

e^{x} = - 6 + 37

f (x) = g (x)

, then

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

ln (e^{x}) = ln (- 6 + 37)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (- 6 + 37)

x = ln (- 6 + 37)

Solve

e^{x} = - 6 - 37 :

No Solution for

x \in R

e^{x} = - 6 - 37

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (- 6 + 37)

x = ln (- 6 + 37)

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)=-6 ?
The general solution for sinh(x)=-6 is x=ln(-6+sqrt(37))