What is the general solution for sinh(x)=3 ?

The general solution for sinh(x)=3 is x=ln(3+sqrt(10))

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sinh(x)=3

Solution

sinh (x) = 3

Solution

x = ln (3 + 10)

Degrees

x = 104.18930 \dots^{\circ}

Solution steps

sinh (x) = 3

Rewrite using trig identities

sinh (x) = 3

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} = 3

\frac{e ^{x} - e ^{- x}}{2} = 3

\frac{e ^{x} - e ^{- x}}{2} = 3 : x = ln (3 + 10)

\frac{e ^{x} - e ^{- x}}{2} = 3

Multiply both sides by

2

\frac{e ^{x} - e ^{- x}}{2} \cdot 2 = 3 \cdot 2

Simplify

e^{x} - e^{- x} = 6

Apply exponent rules

e^{x} - e^{- x} = 6

Apply exponent rule:

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

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

e^{x} - (e^{x})^{- 1} = 6

e^{x} - (e^{x})^{- 1} = 6

Rewrite the equation with

e^{x} = u

u - (u)^{- 1} = 6

Solve

u - u^{- 1} = 6 : u = 3 + 10, u = 3 - 10

u - u^{- 1} = 6

Refine

u - \frac{1}{u} = 6

Multiply both sides by

u

u - \frac{1}{u} = 6

Multiply both sides by

u

uu - \frac{1}{u} u = 6 u

Simplify

uu - \frac{1}{u} u = 6 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 = 6 u

u^{2} - 1 = 6 u

u^{2} - 1 = 6 u

Solve

u^{2} - 1 = 6 u : u = 3 + 10, u = 3 - 10

u^{2} - 1 = 6 u

Move

6 u

to the left side

u^{2} - 1 = 6 u

Subtract

6 u

from both sides

u^{2} - 1 - 6 u = 6 u - 6 u

Simplify

u^{2} - 1 - 6 u = 0

u^{2} - 1 - 6 u = 0

Write in the standard form

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

u^{2} - 6 u - 1 = 0

Solve with the quadratic formula

u^{2} - 6 u - 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 6, c = - 1

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

(- 6)^{2} - 4 \cdot 1 \cdot (- 1) = 210

(- 6)^{2} - 4 \cdot 1 \cdot (- 1)

Apply rule

- (- a) = a

= (- 6)^{2} + 4 \cdot 1 \cdot 1

Apply exponent rule:

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

n

is even

(- 6)^{2} = 6^{2}

= 6^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 6^{2} + 4

6^{2} = 36

= 36 + 4

Add the numbers:

36 + 4 = 40

= 40

Prime factorization of

40 : 2^{3} \cdot 5

40

40

divides by

240 = 20 \cdot 2

= 2 \cdot 20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 5

= 2^{3} \cdot 5

= 2^{3} \cdot 5

Apply exponent rule:

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

= 2^{2} \cdot 2 \cdot 5

Apply radical rule:

= 2^{2} 2 \cdot 5

Apply radical rule:

2^{2} = 2

= 2 2 \cdot 5

Refine

= 210

u_{1, 2} = \frac{- ( - 6 ) \pm 2 10}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 6 ) + 2 10}{2 \cdot 1}, u_{2} = \frac{- ( - 6 ) - 2 10}{2 \cdot 1}

u = \frac{- ( - 6 ) + 2 10}{2 \cdot 1} : 3 + 10

\frac{- ( - 6 ) + 2 10}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{6 + 2 10}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{6 + 2 10}{2}

Factor

6 + 210 : 2 (3 + 10)

6 + 210

Rewrite as

= 2 \cdot 3 + 210

Factor out common term

2

= 2 (3 + 10)

= \frac{2 ( 3 + 10 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 + 10

u = \frac{- ( - 6 ) - 2 10}{2 \cdot 1} : 3 - 10

\frac{- ( - 6 ) - 2 10}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{6 - 2 10}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{6 - 2 10}{2}

Factor

6 - 210 : 2 (3 - 10)

6 - 210

Rewrite as

= 2 \cdot 3 - 210

Factor out common term

2

= 2 (3 - 10)

= \frac{2 ( 3 - 10 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 - 10

The solutions to the quadratic equation are:

u = 3 + 10, u = 3 - 10

u = 3 + 10, u = 3 - 10

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 = 3 + 10, u = 3 - 10

u = 3 + 10, u = 3 - 10

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 3 + 10 : x = ln (3 + 10)

e^{x} = 3 + 10

Apply exponent rules

e^{x} = 3 + 10

f (x) = g (x)

, then

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

ln (e^{x}) = ln (3 + 10)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3 + 10)

x = ln (3 + 10)

Solve

e^{x} = 3 - 10 :

No Solution for

x \in R

e^{x} = 3 - 10

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (3 + 10)

x = ln (3 + 10)

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)=3 ?
The general solution for sinh(x)=3 is x=ln(3+sqrt(10))