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

The general solution for sinh(x)+4=4cosh(x) is x=0,x=ln(5/3)

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sinh(x)+4=4cosh(x)

Solution

sinh (x) + 4 = 4 cosh (x)

Solution

x = 0, x = ln (\frac{5}{3})

Degrees

x = 0^{\circ}, x = 29.26815 \dots^{\circ}

Solution steps

sinh (x) + 4 = 4 cosh (x)

Rewrite using trig identities

sinh (x) + 4 = 4 cosh (x)

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} + 4 = 4 cosh (x)

Use the Hyperbolic identity:

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

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

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

\frac{e ^{x} - e ^{- x}}{2} + 4 = 4 \cdot \frac{e ^{x} + e ^{- x}}{2} : x = 0, x = ln (\frac{5}{3})

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

Multiply both sides by

2

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

Simplify

e^{x} - e^{- x} + 8 = 4 (e^{x} + e^{- x})

Apply exponent rules

e^{x} - e^{- x} + 8 = 4 (e^{x} + e^{- x})

Apply exponent rule:

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

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

e^{x} - (e^{x})^{- 1} + 8 = 4 (e^{x} + (e^{x})^{- 1})

e^{x} - (e^{x})^{- 1} + 8 = 4 (e^{x} + (e^{x})^{- 1})

Rewrite the equation with

e^{x} = u

u - (u)^{- 1} + 8 = 4 (u + (u)^{- 1})

Solve

u - u^{- 1} + 8 = 4 (u + u^{- 1}) : u = 1, u = \frac{5}{3}

u - u^{- 1} + 8 = 4 (u + u^{- 1})

Refine

u - \frac{1}{u} + 8 = 4 (u + \frac{1}{u})

Multiply both sides by

u

u - \frac{1}{u} + 8 = 4 (u + \frac{1}{u})

Multiply both sides by

u

uu - \frac{1}{u} u + 8 u = 4 (u + \frac{1}{u}) u

Simplify

uu - \frac{1}{u} u + 8 u = 4 (u + \frac{1}{u}) 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 = 4 (u + \frac{1}{u}) u

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

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

Expand

4 (u + \frac{1}{u}) u : 4 u^{2} + 4

4 (u + \frac{1}{u}) u

= 4 u (u + \frac{1}{u})

Apply the distributive law:

a (b + c) = ab + a c

a = 4 u, b = u, c = \frac{1}{u}

= 4 uu + 4 u \frac{1}{u}

= 4 uu + 4 \cdot \frac{1}{u} u

Simplify

4 uu + 4 \cdot \frac{1}{u} u : 4 u^{2} + 4

4 uu + 4 \cdot \frac{1}{u} u

4 uu = 4 u^{2}

4 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 4 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 4 u^{2}

4 \cdot \frac{1}{u} u = 4

4 \cdot \frac{1}{u} u

Multiply fractions:

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

= \frac{1 \cdot 4 u}{u}

Cancel the common factor:

u

= 1 \cdot 4

Multiply the numbers:

1 \cdot 4 = 4

= 4

= 4 u^{2} + 4

= 4 u^{2} + 4

u^{2} - 1 + 8 u = 4 u^{2} + 4

Move

1

to the right side

u^{2} - 1 + 8 u = 4 u^{2} + 4

Add

1

to both sides

u^{2} - 1 + 8 u + 1 = 4 u^{2} + 4 + 1

Simplify

u^{2} + 8 u = 4 u^{2} + 5

u^{2} + 8 u = 4 u^{2} + 5

Solve

u^{2} + 8 u = 4 u^{2} + 5 : u = 1, u = \frac{5}{3}

u^{2} + 8 u = 4 u^{2} + 5

Move

5

to the left side

u^{2} + 8 u = 4 u^{2} + 5

Subtract

5

from both sides

u^{2} + 8 u - 5 = 4 u^{2} + 5 - 5

Simplify

u^{2} + 8 u - 5 = 4 u^{2}

u^{2} + 8 u - 5 = 4 u^{2}

Move

4 u^{2}

to the left side

u^{2} + 8 u - 5 = 4 u^{2}

Subtract

4 u^{2}

from both sides

u^{2} + 8 u - 5 - 4 u^{2} = 4 u^{2} - 4 u^{2}

Simplify

- 3 u^{2} + 8 u - 5 = 0

- 3 u^{2} + 8 u - 5 = 0

Solve with the quadratic formula

- 3 u^{2} + 8 u - 5 = 0

Quadratic Equation Formula:

For

a = - 3, b = 8, c = - 5

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) ( - 5 )}{2 ( - 3 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) ( - 5 )}{2 ( - 3 )}

8^{2} - 4 (- 3) (- 5) = 2

8^{2} - 4 (- 3) (- 5)

Apply rule

- (- a) = a

= 8^{2} - 4 \cdot 3 \cdot 5

Multiply the numbers:

4 \cdot 3 \cdot 5 = 60

= 8^{2} - 60

8^{2} = 64

= 64 - 60

Subtract the numbers:

64 - 60 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u_{1, 2} = \frac{- 8 \pm 2}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- 8 + 2}{2 ( - 3 )}, u_{2} = \frac{- 8 - 2}{2 ( - 3 )}

u = \frac{- 8 + 2}{2 ( - 3 )} : 1

\frac{- 8 + 2}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 2}{- 2 \cdot 3}

Add/Subtract the numbers:

- 8 + 2 = - 6

= \frac{- 6}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 6}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- 8 - 2}{2 ( - 3 )} : \frac{5}{3}

\frac{- 8 - 2}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 2}{- 2 \cdot 3}

Subtract the numbers:

- 8 - 2 = - 10

= \frac{- 10}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 10}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{10}{6}

Cancel the common factor:

2

= \frac{5}{3}

The solutions to the quadratic equation are:

u = 1, u = \frac{5}{3}

u = 1, u = \frac{5}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u - u^{- 1} + 8

and compare to zero

u = 0

Take the denominator(s) of

4 (u + u^{- 1})

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = \frac{5}{3}

u = 1, u = \frac{5}{3}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 1 : x = 0

e^{x} = 1

Apply exponent rules

e^{x} = 1

f (x) = g (x)

, then

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

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

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (1)

Simplify

ln (1) : 0

ln (1)

Apply log rule:

lo g_{a} (1) = 0

= 0

x = 0

x = 0

Solve

e^{x} = \frac{5}{3} : x = ln (\frac{5}{3})

e^{x} = \frac{5}{3}

Apply exponent rules

e^{x} = \frac{5}{3}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{5}{3})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{5}{3})

x = ln (\frac{5}{3})

x = 0, x = ln (\frac{5}{3})

x = 0, x = ln (\frac{5}{3})

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)+4=4cosh(x) ?
The general solution for sinh(x)+4=4cosh(x) is x=0,x=ln(5/3)