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

The general solution for 5=sinh(x) is x=ln(5+sqrt(26))

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5=sinh(x)

Solution

5 = sinh (x)

Solution

x = ln (5 + 26)

Degrees

x = 132.49295 \dots^{\circ}

Solution steps

5 = sinh (x)

Switch sides

sinh (x) = 5

Rewrite using trig identities

sinh (x) = 5

Use the Hyperbolic identity:

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

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

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

\frac{e ^{x} - e ^{- x}}{2} = 5 : x = ln (5 + 26)

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

Multiply both sides by

2

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

Simplify

e^{x} - e^{- x} = 10

Apply exponent rules

e^{x} - e^{- x} = 10

Apply exponent rule:

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

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

e^{x} - (e^{x})^{- 1} = 10

e^{x} - (e^{x})^{- 1} = 10

Rewrite the equation with

e^{x} = u

u - (u)^{- 1} = 10

Solve

u - u^{- 1} = 10 : u = 5 + 26, u = 5 - 26

u - u^{- 1} = 10

Refine

u - \frac{1}{u} = 10

Multiply both sides by

u

u - \frac{1}{u} = 10

Multiply both sides by

u

uu - \frac{1}{u} u = 10 u

Simplify

uu - \frac{1}{u} u = 10 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 = 10 u

u^{2} - 1 = 10 u

u^{2} - 1 = 10 u

Solve

u^{2} - 1 = 10 u : u = 5 + 26, u = 5 - 26

u^{2} - 1 = 10 u

Move

10 u

to the left side

u^{2} - 1 = 10 u

Subtract

10 u

from both sides

u^{2} - 1 - 10 u = 10 u - 10 u

Simplify

u^{2} - 1 - 10 u = 0

u^{2} - 1 - 10 u = 0

Write in the standard form

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

u^{2} - 10 u - 1 = 0

Solve with the quadratic formula

u^{2} - 10 u - 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 10, c = - 1

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

(- 10)^{2} - 4 \cdot 1 \cdot (- 1) = 226

(- 10)^{2} - 4 \cdot 1 \cdot (- 1)

Apply rule

- (- a) = a

= (- 10)^{2} + 4 \cdot 1 \cdot 1

Apply exponent rule:

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

n

is even

(- 10)^{2} = 1 0^{2}

= 1 0^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 1 0^{2} + 4

1 0^{2} = 100

= 100 + 4

Add the numbers:

100 + 4 = 104

= 104

Prime factorization of

104 : 2^{3} \cdot 13

104

104

divides by

2104 = 52 \cdot 2

= 2 \cdot 52

52

divides by

252 = 26 \cdot 2

= 2 \cdot 2 \cdot 26

26

divides by

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 13

2, 13

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 13

= 2^{3} \cdot 13

= 2^{3} \cdot 13

Apply exponent rule:

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

= 2^{2} \cdot 2 \cdot 13

Apply radical rule:

= 2^{2} 2 \cdot 13

Apply radical rule:

2^{2} = 2

= 2 2 \cdot 13

Refine

= 226

u_{1, 2} = \frac{- ( - 10 ) \pm 2 26}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 10 ) + 2 26}{2 \cdot 1}, u_{2} = \frac{- ( - 10 ) - 2 26}{2 \cdot 1}

u = \frac{- ( - 10 ) + 2 26}{2 \cdot 1} : 5 + 26

\frac{- ( - 10 ) + 2 26}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{10 + 2 26}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{10 + 2 26}{2}

Factor

10 + 226 : 2 (5 + 26)

10 + 226

Rewrite as

= 2 \cdot 5 + 226

Factor out common term

2

= 2 (5 + 26)

= \frac{2 ( 5 + 26 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 5 + 26

u = \frac{- ( - 10 ) - 2 26}{2 \cdot 1} : 5 - 26

\frac{- ( - 10 ) - 2 26}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{10 - 2 26}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{10 - 2 26}{2}

Factor

10 - 226 : 2 (5 - 26)

10 - 226

Rewrite as

= 2 \cdot 5 - 226

Factor out common term

2

= 2 (5 - 26)

= \frac{2 ( 5 - 26 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 5 - 26

The solutions to the quadratic equation are:

u = 5 + 26, u = 5 - 26

u = 5 + 26, u = 5 - 26

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 = 5 + 26, u = 5 - 26

u = 5 + 26, u = 5 - 26

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 5 + 26 : x = ln (5 + 26)

e^{x} = 5 + 26

Apply exponent rules

e^{x} = 5 + 26

f (x) = g (x)

, then

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

ln (e^{x}) = ln (5 + 26)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (5 + 26)

x = ln (5 + 26)

Solve

e^{x} = 5 - 26 :

No Solution for

x \in R

e^{x} = 5 - 26

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (5 + 26)

x = ln (5 + 26)

Popular Examples

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- 16 cos (2 x) = 0

6sqrt(2)+12cos(x-45)=0

62 + 12 cos (x - 4 5^{\circ}) = 0

(2*9.8*l(1-cos(θ)))=0.018

(2 \cdot 9.8 \cdot l (1 - cos (θ))) = 0.018

1/(tan(x))=0.5

\frac{1}{tan ( x )} = 0.5

tan(θ)=-(16.54)/(34.01)

tan (θ) = - \frac{16.54}{34.01}

Frequently Asked Questions (FAQ)

What is the general solution for 5=sinh(x) ?
The general solution for 5=sinh(x) is x=ln(5+sqrt(26))