What is the general solution for 13/12 =cosh(x/(120)) ?

The general solution for 13/12 =cosh(x/(120)) is x=120ln(3/2)

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13/12 =cosh(x/(120))

Solution

\frac{13}{12} = cosh (\frac{x}{120})

Solution

x = 120 ln (\frac{3}{2})

Degrees

x = 2787.77273 \dots^{\circ}

Solution steps

\frac{13}{12} = cosh (\frac{x}{120})

Switch sides

cosh (\frac{x}{120}) = \frac{13}{12}

Rewrite using trig identities

cosh (\frac{x}{120}) = \frac{13}{12}

Use the Hyperbolic identity:

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

\frac{e ^{\frac{x}{120}} + e ^{- \frac{x}{120}}}{2} = \frac{13}{12}

\frac{e ^{\frac{x}{120}} + e ^{- \frac{x}{120}}}{2} = \frac{13}{12}

\frac{e ^{\frac{x}{120}} + e ^{- \frac{x}{120}}}{2} = \frac{13}{12} : x = 120 ln (\frac{3}{2})

\frac{e ^{\frac{x}{120}} + e ^{- \frac{x}{120}}}{2} = \frac{13}{12}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{\frac{x}{120}} + e^{- \frac{x}{120}}) \cdot 12 = 2 \cdot 13

Simplify

(e^{\frac{x}{120}} + e^{- \frac{x}{120}}) \cdot 12 = 26

Apply exponent rules

(e^{\frac{x}{120}} + e^{- \frac{x}{120}}) \cdot 12 = 26

Apply exponent rule:

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

e^{\frac{x}{120}} = (e^{x})^{0.00833 \dots}, e^{- \frac{x}{120}} = (e^{x})^{- 0.00833 \dots}

((e^{x})^{0.00833 \dots} + (e^{x})^{- 0.00833 \dots}) \cdot 12 = 26

((e^{x})^{0.00833 \dots} + (e^{x})^{- 0.00833 \dots}) \cdot 12 = 26

Rewrite the equation with

e^{x} = u

((u)^{0.00833 \dots} + (u)^{- 0.00833 \dots}) \cdot 12 = 26

Solve

(u^{0.00833 \dots} + u^{- 0.00833 \dots}) \cdot 12 = 26 : u = \frac{3 ^{120}}{2 ^{120}}, u = \frac{2 ^{120}}{3 ^{120}}

(u^{0.00833 \dots} + u^{- 0.00833 \dots}) \cdot 12 = 26

Expand

(u^{0.00833 \dots} + u^{- 0.00833 \dots}) \cdot 12 : 12 u^{0.00833 \dots} + \frac{12}{u ^{0.00833 \dots}}

(u^{0.00833 \dots} + u^{- 0.00833 \dots}) \cdot 12

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= 12 (u^{0.00833 \dots} + \frac{1}{u ^{0.00833 \dots}})

Apply the distributive law:

a (b + c) = ab + a c

a = 12, b = u^{0.00833 \dots}, c = \frac{1}{u ^{0.00833 \dots}}

= 12 u^{0.00833 \dots} + 12 \cdot \frac{1}{u ^{0.00833 \dots}}

12 \cdot \frac{1}{u ^{0.00833 \dots}} = \frac{12}{u ^{0.00833 \dots}}

12 \cdot \frac{1}{u ^{0.00833 \dots}}

Multiply fractions:

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

= \frac{1 \cdot 12}{u ^{0.00833 \dots}}

Multiply the numbers:

1 \cdot 12 = 12

= \frac{12}{u ^{0.00833 \dots}}

= 12 u^{0.00833 \dots} + \frac{12}{u ^{0.00833 \dots}}

12 u^{0.00833 \dots} + \frac{12}{u ^{0.00833 \dots}} = 26

Rewrite the equation with

12 v + \frac{12}{v} = 26

Solve

12 v + \frac{12}{v} = 26 : v = \frac{3}{2}, v = \frac{2}{3}

12 v + \frac{12}{v} = 26

Multiply both sides by

v

12 v + \frac{12}{v} = 26

Multiply both sides by

v

12 vv + \frac{12}{v} v = 26 v

Simplify

12 vv + \frac{12}{v} v = 26 v

Simplify

12 vv : 12 v^{2}

12 vv

Apply exponent rule:

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

vv = v^{1 + 1}

= 12 v^{1 + 1}

Add the numbers:

1 + 1 = 2

= 12 v^{2}

Simplify

\frac{12}{v} v : 12

\frac{12}{v} v

Multiply fractions:

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

= \frac{12 v}{v}

Cancel the common factor:

v

= 12

12 v^{2} + 12 = 26 v

12 v^{2} + 12 = 26 v

12 v^{2} + 12 = 26 v

Solve

12 v^{2} + 12 = 26 v : v = \frac{3}{2}, v = \frac{2}{3}

12 v^{2} + 12 = 26 v

Move

26 v

to the left side

12 v^{2} + 12 = 26 v

Subtract

26 v

from both sides

12 v^{2} + 12 - 26 v = 26 v - 26 v

Simplify

12 v^{2} + 12 - 26 v = 0

12 v^{2} + 12 - 26 v = 0

Write in the standard form

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

12 v^{2} - 26 v + 12 = 0

Solve with the quadratic formula

12 v^{2} - 26 v + 12 = 0

Quadratic Equation Formula:

For

a = 12, b = - 26, c = 12

v_{1, 2} = \frac{- ( - 26 ) \pm ( - 26 ) ^{2} - 4 \cdot 12 \cdot 12}{2 \cdot 12}

v_{1, 2} = \frac{- ( - 26 ) \pm ( - 26 ) ^{2} - 4 \cdot 12 \cdot 12}{2 \cdot 12}

(- 26)^{2} - 4 \cdot 12 \cdot 12 = 10

(- 26)^{2} - 4 \cdot 12 \cdot 12

Apply exponent rule:

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

n

is even

(- 26)^{2} = 2 6^{2}

= 2 6^{2} - 4 \cdot 12 \cdot 12

Multiply the numbers:

4 \cdot 12 \cdot 12 = 576

= 2 6^{2} - 576

2 6^{2} = 676

= 676 - 576

Subtract the numbers:

676 - 576 = 100

= 100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

v_{1, 2} = \frac{- ( - 26 ) \pm 10}{2 \cdot 12}

Separate the solutions

v_{1} = \frac{- ( - 26 ) + 10}{2 \cdot 12}, v_{2} = \frac{- ( - 26 ) - 10}{2 \cdot 12}

v = \frac{- ( - 26 ) + 10}{2 \cdot 12} : \frac{3}{2}

\frac{- ( - 26 ) + 10}{2 \cdot 12}

Apply rule

- (- a) = a

= \frac{26 + 10}{2 \cdot 12}

Add the numbers:

26 + 10 = 36

= \frac{36}{2 \cdot 12}

Multiply the numbers:

2 \cdot 12 = 24

= \frac{36}{24}

Cancel the common factor:

12

= \frac{3}{2}

v = \frac{- ( - 26 ) - 10}{2 \cdot 12} : \frac{2}{3}

\frac{- ( - 26 ) - 10}{2 \cdot 12}

Apply rule

- (- a) = a

= \frac{26 - 10}{2 \cdot 12}

Subtract the numbers:

26 - 10 = 16

= \frac{16}{2 \cdot 12}

Multiply the numbers:

2 \cdot 12 = 24

= \frac{16}{24}

Cancel the common factor:

8

= \frac{2}{3}

The solutions to the quadratic equation are:

v = \frac{3}{2}, v = \frac{2}{3}

v = \frac{3}{2}, v = \frac{2}{3}

Verify Solutions

Find undefined (singularity) points:

v = 0

Take the denominator(s) of

12 v + \frac{12}{v}

and compare to zero

v = 0

The following points are undefined

v = 0

Combine undefined points with solutions:

v = \frac{3}{2}, v = \frac{2}{3}

v = \frac{3}{2}, v = \frac{2}{3}

Substitute back

solve for

u

Solve

Take both sides of the equation to the power of

120 : u = \frac{3 ^{120}}{2 ^{120}}

Expand

Apply radical rule:

= (u^{\frac{1}{120}})^{120}

Apply exponent rule:

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

= u^{\frac{1}{120} \cdot 120}

\frac{1}{120} \cdot 120 = 1

\frac{1}{120} \cdot 120

Multiply fractions:

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

= \frac{1 \cdot 120}{120}

Cancel the common factor:

120

= 1

= u

Expand

(\frac{3}{2})^{120} : \frac{3 ^{120}}{2 ^{120}}

(\frac{3}{2})^{120}

Apply exponent rule:

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

= \frac{3 ^{120}}{2 ^{120}}

u = \frac{3 ^{120}}{2 ^{120}}

u = \frac{3 ^{120}}{2 ^{120}}

Verify Solutions:

u = \frac{3 ^{120}}{2 ^{120}}

True

Check the solutions by plugging them into

Remove the ones that don't agree with the equation.

Plug in

u = \frac{3 ^{120}}{2 ^{120}} :

True

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply radical rule:

assuming

a \geq 0

Apply radical rule:

assuming

a \geq 0

= \frac{3}{2}

\frac{3}{2} = \frac{3}{2}

True

The solution is

u = \frac{3 ^{120}}{2 ^{120}}

Solve

Take both sides of the equation to the power of

120 : u = \frac{2 ^{120}}{3 ^{120}}

Expand

Apply radical rule:

= (u^{\frac{1}{120}})^{120}

Apply exponent rule:

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

= u^{\frac{1}{120} \cdot 120}

\frac{1}{120} \cdot 120 = 1

\frac{1}{120} \cdot 120

Multiply fractions:

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

= \frac{1 \cdot 120}{120}

Cancel the common factor:

120

= 1

= u

Expand

(\frac{2}{3})^{120} : \frac{2 ^{120}}{3 ^{120}}

(\frac{2}{3})^{120}

Apply exponent rule:

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

= \frac{2 ^{120}}{3 ^{120}}

u = \frac{2 ^{120}}{3 ^{120}}

u = \frac{2 ^{120}}{3 ^{120}}

Verify Solutions:

u = \frac{2 ^{120}}{3 ^{120}}

True

Check the solutions by plugging them into

Remove the ones that don't agree with the equation.

Plug in

u = \frac{2 ^{120}}{3 ^{120}} :

True

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply radical rule:

assuming

a \geq 0

Apply radical rule:

assuming

a \geq 0

= \frac{2}{3}

\frac{2}{3} = \frac{2}{3}

True

The solution is

u = \frac{2 ^{120}}{3 ^{120}}

u = \frac{3 ^{120}}{2 ^{120}}, u = \frac{2 ^{120}}{3 ^{120}}

Verify Solutions:

u = \frac{3 ^{120}}{2 ^{120}}

True

, u = \frac{2 ^{120}}{3 ^{120}}

True

Check the solutions by plugging them into

(u^{0.00833 \dots} + u^{- 0.00833 \dots}) 12 = 26

Remove the ones that don't agree with the equation.

Plug in

u = \frac{3 ^{120}}{2 ^{120}} :

True

((\frac{3 ^{120}}{2 ^{120}})^{0.00833 \dots} + (\frac{3 ^{120}}{2 ^{120}})^{- 0.00833 \dots}) \cdot 12 = 26

((\frac{3 ^{120}}{2 ^{120}})^{0.00833 \dots} + (\frac{3 ^{120}}{2 ^{120}})^{- 0.00833 \dots}) \cdot 12 = 26

((\frac{3 ^{120}}{2 ^{120}})^{0.00833 \dots} + (\frac{3 ^{120}}{2 ^{120}})^{- 0.00833 \dots}) \cdot 12

(\frac{3 ^{120}}{2 ^{120}})^{0.00833 \dots} = 1.5

(\frac{3 ^{120}}{2 ^{120}})^{0.00833 \dots}

\frac{3 ^{120}}{2 ^{120}} = 1.35192 E 21

\frac{3 ^{120}}{2 ^{120}}

Convert element to a decimal form

2^{120} = 1.32923 E 36

= \frac{3 ^{120}}{1.32923 E 36}

Convert element to a decimal form

3^{120} = 1.79701 E 57

= \frac{1.79701 E 57}{1.32923 E 36}

Divide the numbers:

\frac{1.79701 E 57}{1.32923 E 36} = 1.35192 E 21

= 1.35192 E 21

= 1.35192 E 2 1^{0.00833 \dots}

1.35192 E 2 1^{0.00833 \dots} = 1.5

= 1.5

(\frac{3 ^{120}}{2 ^{120}})^{- 0.00833 \dots} = 0.66666 \dots

(\frac{3 ^{120}}{2 ^{120}})^{- 0.00833 \dots}

\frac{3 ^{120}}{2 ^{120}} = 1.35192 E 21

\frac{3 ^{120}}{2 ^{120}}

Convert element to a decimal form

2^{120} = 1.32923 E 36

= \frac{3 ^{120}}{1.32923 E 36}

Convert element to a decimal form

3^{120} = 1.79701 E 57

= \frac{1.79701 E 57}{1.32923 E 36}

Divide the numbers:

\frac{1.79701 E 57}{1.32923 E 36} = 1.35192 E 21

= 1.35192 E 21

= 1.35192 E 2 1^{- 0.00833 \dots}

1.35192 E 2 1^{- 0.00833 \dots} = 0.66666 \dots

= 0.66666 \dots

= 12 (0.66666 \dots + 1.5)

Add the numbers:

1.5 + 0.66666 \dots = 2.16666 \dots

= 12 \cdot 2.16666 \dots

Multiply the numbers:

2.16666 \dots \cdot 12 = 26

= 26

26 = 26

True

Plug in

u = \frac{2 ^{120}}{3 ^{120}} :

True

((\frac{2 ^{120}}{3 ^{120}})^{0.00833 \dots} + (\frac{2 ^{120}}{3 ^{120}})^{- 0.00833 \dots}) \cdot 12 = 26

((\frac{2 ^{120}}{3 ^{120}})^{0.00833 \dots} + (\frac{2 ^{120}}{3 ^{120}})^{- 0.00833 \dots}) \cdot 12 = 26

((\frac{2 ^{120}}{3 ^{120}})^{0.00833 \dots} + (\frac{2 ^{120}}{3 ^{120}})^{- 0.00833 \dots}) \cdot 12

(\frac{2 ^{120}}{3 ^{120}})^{0.00833 \dots} = 0.66666 \dots

(\frac{2 ^{120}}{3 ^{120}})^{0.00833 \dots}

\frac{2 ^{120}}{3 ^{120}} = 7.39689 E - 22

\frac{2 ^{120}}{3 ^{120}}

Convert element to a decimal form

3^{120} = 1.79701 E 57

= \frac{2 ^{120}}{1.79701 E 57}

Convert element to a decimal form

2^{120} = 1.32923 E 36

= \frac{1.32923 E 36}{1.79701 E 57}

Divide the numbers:

\frac{1.32923 E 36}{1.79701 E 57} = 7.39689 E - 22

= 7.39689 E - 22

= 7.39689 E - 2 2^{0.00833 \dots}

7.39689 E - 2 2^{0.00833 \dots} = 0.66666 \dots

= 0.66666 \dots

(\frac{2 ^{120}}{3 ^{120}})^{- 0.00833 \dots} = 1.5

(\frac{2 ^{120}}{3 ^{120}})^{- 0.00833 \dots}

\frac{2 ^{120}}{3 ^{120}} = 7.39689 E - 22

\frac{2 ^{120}}{3 ^{120}}

Convert element to a decimal form

3^{120} = 1.79701 E 57

= \frac{2 ^{120}}{1.79701 E 57}

Convert element to a decimal form

2^{120} = 1.32923 E 36

= \frac{1.32923 E 36}{1.79701 E 57}

Divide the numbers:

\frac{1.32923 E 36}{1.79701 E 57} = 7.39689 E - 22

= 7.39689 E - 22

= 7.39689 E - 2 2^{- 0.00833 \dots}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{7.39689 E - 2 2 ^{0.00833 \dots}}

7.39689 E - 2 2^{0.00833 \dots} = 0.66666 \dots

= \frac{1}{0.66666 \dots}

Divide the numbers:

\frac{1}{0.66666 \dots} = 1.5

= 1.5

= 12 (0.66666 \dots + 1.5)

Add the numbers:

0.66666 \dots + 1.5 = 2.16666 \dots

= 12 \cdot 2.16666 \dots

Multiply the numbers:

2.16666 \dots \cdot 12 = 26

= 26

26 = 26

True

The solutions are

u = \frac{3 ^{120}}{2 ^{120}}, u = \frac{2 ^{120}}{3 ^{120}}

u = \frac{3 ^{120}}{2 ^{120}}, u = \frac{2 ^{120}}{3 ^{120}}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{3 ^{120}}{2 ^{120}} : x = 120 ln (\frac{3}{2})

e^{x} = \frac{3 ^{120}}{2 ^{120}}

Apply exponent rules

e^{x} = \frac{3 ^{120}}{2 ^{120}}

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{2 ^{120}} = 2^{- 120}

e^{x} = 3^{120} \cdot 2^{- 120}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (3^{120} \cdot 2^{- 120})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3^{120} \cdot 2^{- 120})

Simplify

ln (3^{120} \cdot 2^{- 120}) : 120 ln (\frac{3}{2})

ln (3^{120} \cdot 2^{- 120})

Multiply

3^{120} \cdot 2^{- 120} : \frac{3 ^{120}}{2 ^{120}}

3^{120} \cdot 2^{- 120}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

2^{- 120} = \frac{1}{2 ^{120}}

= 3^{120} \cdot \frac{1}{2 ^{120}}

Multiply fractions:

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

= \frac{1 \cdot 3 ^{120}}{2 ^{120}}

Multiply:

1 \cdot 3^{120} = 3^{120}

= \frac{3 ^{120}}{2 ^{120}}

= ln (\frac{3 ^{120}}{2 ^{120}})

Combine same powers :

\frac{x ^{n}}{y ^{n}} = (\frac{x}{y})^{n}

= ln ((\frac{3}{2})^{120})

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= 120 ln (\frac{3}{2})

x = 120 ln (\frac{3}{2})

x = 120 ln (\frac{3}{2})

Solve

e^{x} = \frac{2 ^{120}}{3 ^{120}} :

No Solution for

x \in R

e^{x} = \frac{2 ^{120}}{3 ^{120}}

Apply exponent rules

e^{x} = \frac{2 ^{120}}{3 ^{120}}

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{3 ^{120}} = 3^{- 120}

e^{x} = 2^{120} \cdot 3^{- 120}

e^{x} = 2^{120} \cdot 3^{- 120}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = 120 ln (\frac{3}{2})

x = 120 ln (\frac{3}{2})

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Frequently Asked Questions (FAQ)

What is the general solution for 13/12 =cosh(x/(120)) ?
The general solution for 13/12 =cosh(x/(120)) is x=120ln(3/2)