What is the general solution for sinh(x)= 5/12 ,cosh(x) ?

The general solution for sinh(x)= 5/12 ,cosh(x) is x=ln(3/2)

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Popular Trigonometry >

sinh(x)= 5/12 ,cosh(x)

Solution

sinh (x) = \frac{5}{12}, cosh (x)

Solution

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

Degrees

x = 23.23143 \dots^{\circ}

Solution steps

sinh (x) = \frac{5}{12}, cosh (x)

Rewrite using trig identities

sinh (x) = \frac{5}{12}

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{12}

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{12}

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{12} : x = ln (\frac{3}{2})

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{12}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 12 = 2 \cdot 5

Simplify

(e^{x} - e^{- x}) \cdot 12 = 10

Apply exponent rules

(e^{x} - e^{- x}) \cdot 12 = 10

Apply exponent rule:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 12 = 10

(e^{x} - (e^{x})^{- 1}) \cdot 12 = 10

Rewrite the equation with

e^{x} = u

(u - (u)^{- 1}) \cdot 12 = 10

Solve

(u - u^{- 1}) \cdot 12 = 10 : u = \frac{3}{2}, u = - \frac{2}{3}

(u - u^{- 1}) \cdot 12 = 10

Refine

(u - \frac{1}{u}) \cdot 12 = 10

Simplify

(u - \frac{1}{u}) \cdot 12 : 12 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 12

Apply the commutative law:

(u - \frac{1}{u}) \cdot 12 = 12 (u - \frac{1}{u})

12 (u - \frac{1}{u})

12 (u - \frac{1}{u}) = 10

Expand

12 (u - \frac{1}{u}) : 12 u - \frac{12}{u}

12 (u - \frac{1}{u})

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

12 \cdot \frac{1}{u}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 12 = 12

= \frac{12}{u}

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

12 u - \frac{12}{u} = 10

Multiply both sides by

u

12 u - \frac{12}{u} = 10

Multiply both sides by

u

12 uu - \frac{12}{u} u = 10 u

Simplify

12 uu - \frac{12}{u} u = 10 u

Simplify

12 uu : 12 u^{2}

12 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 12 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 12 u^{2}

Simplify

- \frac{12}{u} u : - 12

- \frac{12}{u} u

Multiply fractions:

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

= - \frac{12 u}{u}

Cancel the common factor:

u

= - 12

12 u^{2} - 12 = 10 u

12 u^{2} - 12 = 10 u

12 u^{2} - 12 = 10 u

Solve

12 u^{2} - 12 = 10 u : u = \frac{3}{2}, u = - \frac{2}{3}

12 u^{2} - 12 = 10 u

Move

10 u

to the left side

12 u^{2} - 12 = 10 u

Subtract

10 u

from both sides

12 u^{2} - 12 - 10 u = 10 u - 10 u

Simplify

12 u^{2} - 12 - 10 u = 0

12 u^{2} - 12 - 10 u = 0

Write in the standard form

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

12 u^{2} - 10 u - 12 = 0

Solve with the quadratic formula

12 u^{2} - 10 u - 12 = 0

Quadratic Equation Formula:

For

a = 12, b = - 10, c = - 12

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 12 ( - 12 )}{2 \cdot 12}

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 12 ( - 12 )}{2 \cdot 12}

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

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

= 1 0^{2} + 4 \cdot 12 \cdot 12

Multiply the numbers:

4 \cdot 12 \cdot 12 = 576

= 1 0^{2} + 576

1 0^{2} = 100

= 100 + 576

Add the numbers:

100 + 576 = 676

= 676

Factor the number:

676 = 2 6^{2}

= 2 6^{2}

Apply radical rule:

2 6^{2} = 26

= 26

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

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Add the numbers:

10 + 26 = 36

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

Multiply the numbers:

2 \cdot 12 = 24

= \frac{36}{24}

Cancel the common factor:

12

= \frac{3}{2}

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

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

Apply rule

- (- a) = a

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

Subtract the numbers:

10 - 26 = - 16

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

Multiply the numbers:

2 \cdot 12 = 24

= \frac{- 16}{24}

Apply the fraction rule:

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

= - \frac{16}{24}

Cancel the common factor:

8

= - \frac{2}{3}

The solutions to the quadratic equation are:

u = \frac{3}{2}, u = - \frac{2}{3}

u = \frac{3}{2}, u = - \frac{2}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u - u^{- 1}) 12

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{3}{2}, u = - \frac{2}{3}

u = \frac{3}{2}, u = - \frac{2}{3}

Substitute back

u = e^{x},

solve for

x

Solve

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

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

Apply exponent rules

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

f (x) = g (x)

, then

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

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

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

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

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

Solve

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

No Solution for

x \in R

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

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

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

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)= 5/12 ,cosh(x) ?
The general solution for sinh(x)= 5/12 ,cosh(x) is x=ln(3/2)