What is the general solution for tanh(mL)=0.99 ?

The general solution for tanh(mL)=0.99 is L=(ln(\frac{1.99)/(0.01))}{2m}

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tanh(mL)=0.99

Solution

tanh (m L) = 0.99

Solution

L = \frac{ln ( \frac{1.99}{0.01} )}{2 m}

Solution steps

tanh (m L) = 0.99

Rewrite using trig identities

tanh (m L) = 0.99

Use the Hyperbolic identity:

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

\frac{e ^{m L} - e ^{- m L}}{e ^{m L} + e ^{- m L}} = 0.99

\frac{e ^{m L} - e ^{- m L}}{e ^{m L} + e ^{- m L}} = 0.99

\frac{e ^{m L} - e ^{- m L}}{e ^{m L} + e ^{- m L}} = 0.99 : L = \frac{ln ( \frac{1.99}{0.01} )}{2 m} {m < 0 or m > 0}

\frac{e ^{m L} - e ^{- m L}}{e ^{m L} + e ^{- m L}} = 0.99

Multiply both sides by

e^{m L} + e^{- m L}

\frac{e ^{m L} - e ^{- m L}}{e ^{m L} + e ^{- m L}} (e^{m L} + e^{- m L}) = 0.99 (e^{m L} + e^{- m L})

Simplify

e^{m L} - e^{- m L} = 0.99 (e^{m L} + e^{- m L})

Subtract

0.99 (e^{m L} + e^{- m L})

from both sides

e^{m L} - e^{- m L} - 0.99 (e^{m L} + e^{- m L}) = 0.99 (e^{m L} + e^{- m L}) - 0.99 (e^{m L} + e^{- m L})

Simplify

e^{m L} - e^{- m L} - 0.99 (e^{m L} + e^{- m L}) = 0

Factor

e^{m L} - e^{- m L} - 0.99 (e^{m L} + e^{- m L}) : e^{- m L} (0.01 e^{m L} + 1.99) (0.01 e^{m L} - 1.99)

e^{m L} - e^{- m L} - 0.99 (e^{m L} + e^{- m L})

Factor

e^{m L} + e^{- m L} : e^{- m L} (e^{2 m L} + 1)

e^{m L} + e^{- m L}

Apply exponent rule:

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

e^{m L} = e^{- m L} e^{2 m L}

= e^{- m L} e^{2 m L} + e^{- m L}

Factor out common term

e^{- m L}

= e^{- m L} (e^{2 m L} + 1)

= e^{m L} - e^{- m L} - 0.99 e^{- m L} (e^{2 m L} + 1)

Apply exponent rule:

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

e^{m L} = e^{- m L} e^{2 m L}

= e^{- m L} e^{2 m L} - e^{- m L} - 0.99 e^{- m L} (e^{2 m L} + 1)

Factor out common term

e^{- m L}

= e^{- m L} (e^{2 m L} - 1 - 0.99 (e^{2 m L} + 1))

Factor

e^{2 m L} - 0.99 (e^{2 m L} + 1) - 1 : (0.01 e^{m L} + 1.99) (0.01 e^{m L} - 1.99)

e^{2 m L} - 1 - 0.99 (e^{2 m L} + 1)

Expand

- 0.99 (e^{2 m L} + 1) : - 0.99 e^{2 m L} - 0.99

- 0.99 (e^{2 m L} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = - 0.99, b = e^{2 m L}, c = 1

= - 0.99 e^{2 m L} + (- 0.99) \cdot 1

Apply minus-plus rules

+ (- a) = - a

= - 0.99 e^{2 m L} - 1 \cdot 0.99

Multiply the numbers:

1 \cdot 0.99 = 0.99

= - 0.99 e^{2 m L} - 0.99

= e^{2 m L} - 1 - 0.99 e^{2 m L} - 0.99

Simplify

e^{2 m L} - 1 - 0.99 e^{2 m L} - 0.99 : 0.01 e^{2 m L} - 1.99

e^{2 m L} - 1 - 0.99 e^{2 m L} - 0.99

Group like terms

= e^{2 m L} - 0.99 e^{2 m L} - 1 - 0.99

Add similar elements:

e^{2 m L} - 0.99 e^{2 m L} = 0.01 e^{2 m L}

= 0.01 e^{2 m L} - 1 - 0.99

Subtract the numbers:

- 1 - 0.99 = - 1.99

= 0.01 e^{2 m L} - 1.99

= 0.01 e^{2 m L} - 1.99

Apply exponent rule:

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

e^{2 m L} = (e^{m L})^{2}

= 0.01 (e^{m L})^{2} - 1.99

Rewrite

0.01 (e^{m L})^{2} - 1.99

(0.01 e^{m L})^{2} - (1.99)^{2}

0.01 (e^{m L})^{2} - 1.99

Apply radical rule:

a = (a)^{2}

0.01 = (0.01)^{2}

= (0.01)^{2} (e^{m L})^{2} - 1.99

Apply radical rule:

a = (a)^{2}

1.99 = (1.99)^{2}

= (0.01)^{2} (e^{m L})^{2} - (1.99)^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(0.01)^{2} (e^{m L})^{2} = (0.01 e^{m L})^{2}

= (0.01 e^{m L})^{2} - (1.99)^{2}

= (0.01 e^{m L})^{2} - (1.99)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(0.01 e^{m L})^{2} - (1.99)^{2} = (0.01 e^{m L} + 1.99) (0.01 e^{m L} - 1.99)

= (0.01 e^{m L} + 1.99) (0.01 e^{m L} - 1.99)

= e^{- m L} (0.01 e^{m L} + 1.99) (0.01 e^{m L} - 1.99)

e^{- m L} (0.01 e^{m L} + 1.99) (0.01 e^{m L} - 1.99) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

e^{- m L} = 0 or 0.01 e^{m L} + 1.99 = 0 or 0.01 e^{m L} - 1.99 = 0

Solve

e^{- m L} = 0 :

No Solution for

L \in R

e^{- m L} = 0

a^{f (L)}

cannot be zero or negative for

L \in R

No Solution for L \in R

Solve

0.01 e^{m L} + 1.99 = 0 :

No Solution for

L \in R

0.01 e^{m L} + 1.99 = 0

Subtract

1.99

from both sides

0.01 e^{m L} + 1.99 - 1.99 = 0 - 1.99

Simplify

0.01 e^{m L} = - 1.99

Divide both sides by

0.01

0.01 e^{m L} = - 1.99

Divide both sides by

0.01

\frac{0.01 e ^{m L}}{0.01} = \frac{- 1.99}{0.01}

Simplify

e^{m L} = - \frac{1.99}{0.01}

e^{m L} = - \frac{1.99}{0.01}

Simplify

e^{m L} = - \frac{1.99}{0.01}

a^{f (L)}

cannot be zero or negative for

L \in R

No Solution for L \in R

Solve

0.01 e^{m L} - 1.99 = 0 : L = \frac{ln ( \frac{1.99}{0.01} )}{2 m}

0.01 e^{m L} - 1.99 = 0

Add

1.99

to both sides

0.01 e^{m L} - 1.99 + 1.99 = 0 + 1.99

Simplify

0.01 e^{m L} = 1.99

Divide both sides by

0.01

0.01 e^{m L} = 1.99

Divide both sides by

0.01

\frac{0.01 e ^{m L}}{0.01} = \frac{1.99}{0.01}

Simplify

e^{m L} = \frac{1.99}{0.01}

e^{m L} = \frac{1.99}{0.01}

Simplify

e^{m L} = \frac{1.99}{0.01}

Apply exponent rules

e^{m L} = \frac{1.99}{0.01}

Apply exponent rule:

a = a^{\frac{1}{2}}

\frac{1.99}{0.01} = (\frac{1.99}{0.01})^{\frac{1}{2}}

e^{m L} = (\frac{1.99}{0.01})^{\frac{1}{2}}

f (x) = g (x)

, then

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

ln (e^{m L}) = ln ((\frac{1.99}{0.01})^{\frac{1}{2}})

Apply log rule:

ln (e^{a}) = a

ln (e^{m L}) = m L

m L = ln ((\frac{1.99}{0.01})^{\frac{1}{2}})

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln ((\frac{1.99}{0.01})^{\frac{1}{2}}) = \frac{1}{2} ln (\frac{1.99}{0.01})

m L = \frac{1}{2} ln (\frac{1.99}{0.01})

m L = \frac{1}{2} ln (\frac{1.99}{0.01})

Solve

m L = \frac{1}{2} ln (\frac{1.99}{0.01}) : L = \frac{ln ( \frac{1.99}{0.01} )}{2 m}

m L = \frac{1}{2} ln (\frac{1.99}{0.01})

Divide both sides by

m

m L = \frac{1}{2} ln (\frac{1.99}{0.01})

Divide both sides by

m

\frac{m L}{m} = \frac{\frac{1}{2} ln ( \frac{1.99}{0.01} )}{m}

Simplify

\frac{m L}{m} = \frac{\frac{1}{2} ln ( \frac{1.99}{0.01} )}{m}

Simplify

\frac{m L}{m} : L

\frac{m L}{m}

Cancel the common factor:

m

= L

Simplify

\frac{\frac{1}{2} ln ( \frac{1.99}{0.01} )}{m} : \frac{ln ( \frac{1.99}{0.01} )}{2 m}

\frac{\frac{1}{2} ln ( \frac{1.99}{0.01} )}{m}

Multiply

\frac{1}{2} ln (\frac{1.99}{0.01}) : \frac{ln ( \frac{1.99}{0.01} )}{2}

\frac{1}{2} ln (\frac{1.99}{0.01})

Multiply fractions:

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

= \frac{1 \cdot ln ( \frac{1.99}{0.01} )}{2}

Multiply:

1 \cdot ln (\frac{1.99}{0.01}) = ln (\frac{1.99}{0.01})

= \frac{ln ( \frac{1.99}{0.01} )}{2}

= \frac{\frac{l n ( \frac{1.99}{0.01} )}{2}}{m}

Apply the fraction rule:

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

= \frac{ln ( \frac{1.99}{0.01} )}{2 m}

L = \frac{ln ( \frac{1.99}{0.01} )}{2 m}

L = \frac{ln ( \frac{1.99}{0.01} )}{2 m}

L = \frac{ln ( \frac{1.99}{0.01} )}{2 m}

L = \frac{ln ( \frac{1.99}{0.01} )}{2 m}

Verify Solutions:

L = \frac{ln ( \frac{1.99}{0.01} )}{2 m} {m < 0 or m > 0}

Check the solutions by plugging them into

\frac{e ^{m L} - e ^{- m L}}{e ^{m L} + e ^{- m L}} = 0.99

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

Plug in

L = \frac{ln ( \frac{1.99}{0.01} )}{2 m} : m < 0 or m > 0

\frac{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} - e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}}{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} + e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}} = 0.99

\frac{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} - e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}}{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} + e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}} = 0.99

\frac{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} - e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}}{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} + e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}}

Remove parentheses:

(a) = a

= \frac{e ^{m \frac{l n ( \frac{1.99}{0.01} )}{2 m}} - e ^{- m \frac{l n ( \frac{1.99}{0.01} )}{2 m}}}{e ^{m \frac{l n ( \frac{1.99}{0.01} )}{2 m}} + e ^{- m \frac{l n ( \frac{1.99}{0.01} )}{2 m}}}

Multiply

m \frac{ln ( \frac{1.99}{0.01} )}{2 m} : 2.64665 \dots

m \frac{ln ( \frac{1.99}{0.01} )}{2 m}

Multiply fractions:

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

= \frac{ln ( \frac{1.99}{0.01} ) m}{2 m}

Cancel the common factor:

m

= \frac{ln ( \frac{1.99}{0.01} )}{2}

Convert element to a decimal form

\frac{1}{2} = 0.5

= 0.5 ln (\frac{1.99}{0.01})

Divide the numbers:

\frac{1.99}{0.01} = 199

= 0.5 ln (199)

Simplify

ln (199) : 5.29330 \dots

ln (199)

Refine to a decimal form

= 5.29330 \dots

= 0.5 \cdot 5.29330 \dots

Multiply the numbers:

0.5 \cdot 5.29330 \dots = 2.64665 \dots

= 2.64665 \dots

= \frac{e ^{m \frac{l n ( \frac{1.99}{0.01} )}{2 m}} - e ^{- m \frac{l n ( \frac{1.99}{0.01} )}{2 m}}}{e ^{2.64665 \dots} + e ^{- m \frac{l n ( \frac{1.99}{0.01} )}{2 m}}}

Multiply

- m \frac{ln ( \frac{1.99}{0.01} )}{2 m} : - 2.64665 \dots

- m \frac{ln ( \frac{1.99}{0.01} )}{2 m}

Multiply fractions:

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

= - \frac{ln ( \frac{1.99}{0.01} ) m}{2 m}

Cancel the common factor:

m

= - \frac{ln ( \frac{1.99}{0.01} )}{2}

Convert element to a decimal form

\frac{1}{2} = 0.5

= - 0.5 ln (\frac{1.99}{0.01})

Divide the numbers:

\frac{1.99}{0.01} = 199

= - 0.5 ln (199)

Simplify

ln (199) : 5.29330 \dots

ln (199)

Refine to a decimal form

= 5.29330 \dots

= - 0.5 \cdot 5.29330 \dots

Multiply the numbers:

0.5 \cdot 5.29330 \dots = 2.64665 \dots

= - 2.64665 \dots

= \frac{e ^{m \frac{l n ( \frac{1.99}{0.01} )}{2 m}} - e ^{- m \frac{l n ( \frac{1.99}{0.01} )}{2 m}}}{e ^{2.64665 \dots} + e ^{- 2.64665 \dots}}

Multiply

m \frac{ln ( \frac{1.99}{0.01} )}{2 m} : 2.64665 \dots

m \frac{ln ( \frac{1.99}{0.01} )}{2 m}

Multiply fractions:

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

= \frac{ln ( \frac{1.99}{0.01} ) m}{2 m}

Cancel the common factor:

m

= \frac{ln ( \frac{1.99}{0.01} )}{2}

Convert element to a decimal form

\frac{1}{2} = 0.5

= 0.5 ln (\frac{1.99}{0.01})

Divide the numbers:

\frac{1.99}{0.01} = 199

= 0.5 ln (199)

Simplify

ln (199) : 5.29330 \dots

ln (199)

Refine to a decimal form

= 5.29330 \dots

= 0.5 \cdot 5.29330 \dots

Multiply the numbers:

0.5 \cdot 5.29330 \dots = 2.64665 \dots

= 2.64665 \dots

= \frac{e ^{2.64665 \dots} - e ^{- m \frac{l n ( \frac{1.99}{0.01} )}{2 m}}}{e ^{2.64665 \dots} + e ^{- 2.64665 \dots}}

Multiply

- m \frac{ln ( \frac{1.99}{0.01} )}{2 m} : - 2.64665 \dots

- m \frac{ln ( \frac{1.99}{0.01} )}{2 m}

Multiply fractions:

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

= - \frac{ln ( \frac{1.99}{0.01} ) m}{2 m}

Cancel the common factor:

m

= - \frac{ln ( \frac{1.99}{0.01} )}{2}

Convert element to a decimal form

\frac{1}{2} = 0.5

= - 0.5 ln (\frac{1.99}{0.01})

Divide the numbers:

\frac{1.99}{0.01} = 199

= - 0.5 ln (199)

Simplify

ln (199) : 5.29330 \dots

ln (199)

Refine to a decimal form

= 5.29330 \dots

= - 0.5 \cdot 5.29330 \dots

Multiply the numbers:

0.5 \cdot 5.29330 \dots = 2.64665 \dots

= - 2.64665 \dots

= \frac{e ^{2.64665 \dots} - e ^{- 2.64665 \dots}}{e ^{2.64665 \dots} + e ^{- 2.64665 \dots}}

Simplify

\frac{e ^{2.64665 \dots} - e ^{- 2.64665 \dots}}{e ^{2.64665 \dots} + e ^{- 2.64665 \dots}}

Apply exponent rule:

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

e^{- 2.64665 \dots} = \frac{1}{e ^{2.64665 \dots}}

= \frac{e ^{2.64665 \dots} - e ^{- 2.64665 \dots}}{e ^{2.64665 \dots} + \frac{1}{e ^{2.64665 \dots}}}

Apply exponent rule:

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

e^{- 2.64665 \dots} = \frac{1}{e ^{2.64665 \dots}}

= \frac{e ^{2.64665 \dots} - \frac{1}{e ^{2.64665 \dots}}}{e ^{2.64665 \dots} + \frac{1}{e ^{2.64665 \dots}}}

Join

e^{2.64665 \dots} + \frac{1}{e ^{2.64665 \dots}} : 14.17762 \dots

e^{2.64665 \dots} + \frac{1}{e ^{2.64665 \dots}}

Convert element to fraction:

e^{2.64665 \dots} = \frac{e ^{2.64665 \dots} e ^{2.64665 \dots}}{e ^{2.64665 \dots}}

= \frac{e ^{2.64665 \dots} e ^{2.64665 \dots}}{e ^{2.64665 \dots}} + \frac{1}{e ^{2.64665 \dots}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{2.64665 \dots} e ^{2.64665 \dots} + 1}{e ^{2.64665 \dots}}

e^{2.64665 \dots} e^{2.64665 \dots} + 1 = e^{5.29330 \dots} + 1

e^{2.64665 \dots} e^{2.64665 \dots} + 1

e^{2.64665 \dots} e^{2.64665 \dots} = e^{5.29330 \dots}

e^{2.64665 \dots} e^{2.64665 \dots}

Apply exponent rule:

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

e^{2.64665 \dots} e^{2.64665 \dots} = e^{2.64665 \dots + 2.64665 \dots}

= e^{2.64665 \dots + 2.64665 \dots}

Add the numbers:

2.64665 \dots + 2.64665 \dots = 5.29330 \dots

= e^{5.29330 \dots}

= e^{5.29330 \dots} + 1

= \frac{e ^{5.29330 \dots} + 1}{e ^{2.64665 \dots}}

e^{5.29330 \dots} = 198.99999 \dots

= \frac{198.99999 \dots + 1}{e ^{2.64665 \dots}}

Add the numbers:

198.99999 \dots + 1 = 199.99999 \dots

= \frac{199.99999 \dots}{e ^{2.64665 \dots}}

e^{2.64665 \dots} = 14.10673 \dots

= \frac{199.99999 \dots}{14.10673 \dots}

Divide the numbers:

\frac{199.99999 \dots}{14.10673 \dots} = 14.17762 \dots

= 14.17762 \dots

= \frac{e ^{2.64665 \dots} - \frac{1}{e ^{2.64665 \dots}}}{14.17762 \dots}

Join

e^{2.64665 \dots} - \frac{1}{e ^{2.64665 \dots}} : 14.03584 \dots

e^{2.64665 \dots} - \frac{1}{e ^{2.64665 \dots}}

Convert element to fraction:

e^{2.64665 \dots} = \frac{e ^{2.64665 \dots} e ^{2.64665 \dots}}{e ^{2.64665 \dots}}

= \frac{e ^{2.64665 \dots} e ^{2.64665 \dots}}{e ^{2.64665 \dots}} - \frac{1}{e ^{2.64665 \dots}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{2.64665 \dots} e ^{2.64665 \dots} - 1}{e ^{2.64665 \dots}}

e^{2.64665 \dots} e^{2.64665 \dots} - 1 = e^{5.29330 \dots} - 1

e^{2.64665 \dots} e^{2.64665 \dots} - 1

e^{2.64665 \dots} e^{2.64665 \dots} = e^{5.29330 \dots}

e^{2.64665 \dots} e^{2.64665 \dots}

Apply exponent rule:

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

e^{2.64665 \dots} e^{2.64665 \dots} = e^{2.64665 \dots + 2.64665 \dots}

= e^{2.64665 \dots + 2.64665 \dots}

Add the numbers:

2.64665 \dots + 2.64665 \dots = 5.29330 \dots

= e^{5.29330 \dots}

= e^{5.29330 \dots} - 1

= \frac{e ^{5.29330 \dots} - 1}{e ^{2.64665 \dots}}

e^{5.29330 \dots} = 198.99999 \dots

= \frac{198.99999 \dots - 1}{e ^{2.64665 \dots}}

Subtract the numbers:

198.99999 \dots - 1 = 197.99999 \dots

= \frac{197.99999 \dots}{e ^{2.64665 \dots}}

e^{2.64665 \dots} = 14.10673 \dots

= \frac{197.99999 \dots}{14.10673 \dots}

Divide the numbers:

\frac{197.99999 \dots}{14.10673 \dots} = 14.03584 \dots

= 14.03584 \dots

= \frac{14.03584 \dots}{14.17762 \dots}

Divide the numbers:

\frac{14.03584 \dots}{14.17762 \dots} = 0.99

= 0.99

= 0.99

0.99 = 0.99

Domain of

\frac{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} - e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}}{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} + e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}} : m < 0 or m > 0

Domain definition

Find undefined (singularity) points:

m = 0

\frac{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} - e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}}{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} + e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}}

Take the denominator(s) of

\frac{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} - e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}}{e ^{m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})} + e ^{- m (\frac{l n ( \frac{1.99}{0.01} )}{2 m})}}

and compare to zero

Solve

2 m = 0 : m = 0

2 m = 0

Divide both sides by

2

2 m = 0

Divide both sides by

2

\frac{2 m}{2} = \frac{0}{2}

Simplify

m = 0

m = 0

The following points are undefined

m = 0

The function domain

m < 0 or m > 0

m < 0 or m > 0

The solution is

L = \frac{ln ( \frac{1.99}{0.01} )}{2 m} {m < 0 or m > 0}

L = \frac{ln ( \frac{1.99}{0.01} )}{2 m}

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csc^{3} (x) - 4 csc (x) = 3 cot^{2} (x)

Frequently Asked Questions (FAQ)

What is the general solution for tanh(mL)=0.99 ?
The general solution for tanh(mL)=0.99 is L=(ln(\frac{1.99)/(0.01))}{2m}