What is the general solution for sinh(x40)= 30/40 ?

The general solution for sinh(x40)= 30/40 is x= 1/40 ln(2)

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sinh(x40)= 30/40

Solution

sinh (x 40) = \frac{30}{40}

Solution

x = \frac{1}{40} ln (2)

Degrees

x = 0.99286 \dots^{\circ}

Solution steps

sinh (x \cdot 40) = \frac{30}{40}

Rewrite using trig identities

sinh (x \cdot 40) = \frac{30}{40}

Use the Hyperbolic identity:

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

\frac{e ^{x \cdot 40} - e ^{- x \cdot 40}}{2} = \frac{30}{40}

\frac{e ^{x \cdot 40} - e ^{- x \cdot 40}}{2} = \frac{30}{40}

\frac{e ^{x \cdot 40} - e ^{- x \cdot 40}}{2} = \frac{30}{40} : x = \frac{1}{40} ln (2)

\frac{e ^{x \cdot 40} - e ^{- x \cdot 40}}{2} = \frac{30}{40}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{x \cdot 40} - e^{- x \cdot 40}) \cdot 40 = 2 \cdot 30

Simplify

(e^{x \cdot 40} - e^{- x \cdot 40}) \cdot 40 = 60

Apply exponent rules

(e^{x \cdot 40} - e^{- x \cdot 40}) \cdot 40 = 60

Apply exponent rule:

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

e^{x 40} = (e^{x})^{40}, e^{- x 40} = (e^{x})^{- 40}

((e^{x})^{40} - (e^{x})^{- 40}) \cdot 40 = 60

((e^{x})^{40} - (e^{x})^{- 40}) \cdot 40 = 60

Rewrite the equation with

e^{x} = u

((u)^{40} - (u)^{- 40}) \cdot 40 = 60

Solve

(u^{40} - u^{- 40}) \cdot 40 = 60

Refine

(u^{40} - \frac{1}{u ^{40}}) \cdot 40 = 60

Simplify

(u^{40} - \frac{1}{u ^{40}}) \cdot 40 : 40 (u^{40} - \frac{1}{u ^{40}})

(u^{40} - \frac{1}{u ^{40}}) \cdot 40

Apply the commutative law:

(u^{40} - \frac{1}{u ^{40}}) \cdot 40 = 40 (u^{40} - \frac{1}{u ^{40}})

40 (u^{40} - \frac{1}{u ^{40}})

40 (u^{40} - \frac{1}{u ^{40}}) = 60

Expand

40 (u^{40} - \frac{1}{u ^{40}}) : 40 u^{40} - \frac{40}{u ^{40}}

40 (u^{40} - \frac{1}{u ^{40}})

Apply the distributive law:

a (b - c) = ab - a c

a = 40, b = u^{40}, c = \frac{1}{u ^{40}}

= 40 u^{40} - 40 \cdot \frac{1}{u ^{40}}

40 \cdot \frac{1}{u ^{40}} = \frac{40}{u ^{40}}

40 \cdot \frac{1}{u ^{40}}

Multiply fractions:

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

= \frac{1 \cdot 40}{u ^{40}}

Multiply the numbers:

1 \cdot 40 = 40

= \frac{40}{u ^{40}}

= 40 u^{40} - \frac{40}{u ^{40}}

40 u^{40} - \frac{40}{u ^{40}} = 60

Multiply both sides by

u^{40}

40 u^{40} - \frac{40}{u ^{40}} = 60

Multiply both sides by

u^{40}

40 u^{40} u^{40} - \frac{40}{u ^{40}} u^{40} = 60 u^{40}

Simplify

40 u^{40} u^{40} - \frac{40}{u ^{40}} u^{40} = 60 u^{40}

Simplify

40 u^{40} u^{40} : 40 u^{80}

40 u^{40} u^{40}

Apply exponent rule:

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

u^{40} u^{40} = u^{40 + 40}

= 40 u^{40 + 40}

Add the numbers:

40 + 40 = 80

= 40 u^{80}

Simplify

- \frac{40}{u ^{40}} u^{40} : - 40

- \frac{40}{u ^{40}} u^{40}

Multiply fractions:

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

= - \frac{40 u ^{40}}{u ^{40}}

Cancel the common factor:

u^{40}

= - 40

40 u^{80} - 40 = 60 u^{40}

40 u^{80} - 40 = 60 u^{40}

40 u^{80} - 40 = 60 u^{40}

Solve

40 u^{80} - 40 = 60 u^{40}

Move

60 u^{40}

to the left side

40 u^{80} - 40 = 60 u^{40}

Subtract

60 u^{40}

from both sides

40 u^{80} - 40 - 60 u^{40} = 60 u^{40} - 60 u^{40}

Simplify

40 u^{80} - 40 - 60 u^{40} = 0

40 u^{80} - 40 - 60 u^{40} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

40 u^{80} - 60 u^{40} - 40 = 0

Rewrite the equation with

v = u^{2}, v^{20} = u^{40}

and

v^{40} = u^{80}

40 v^{40} - 60 v^{20} - 40 = 0

Solve

40 v^{40} - 60 v^{20} - 40 = 0

Rewrite the equation with

u = v^{2}, u^{10} = v^{20}

and

u^{20} = v^{40}

40 u^{20} - 60 u^{10} - 40 = 0

Solve

40 u^{20} - 60 u^{10} - 40 = 0

Rewrite the equation with

v = u^{2}, v^{5} = u^{10}

and

v^{10} = u^{20}

40 v^{10} - 60 v^{5} - 40 = 0

Solve

40 v^{10} - 60 v^{5} - 40 = 0

Rewrite the equation with

u = v^{5}

and

u^{2} = v^{10}

40 u^{2} - 60 u - 40 = 0

Solve

40 u^{2} - 60 u - 40 = 0 : u = 2, u = - \frac{1}{2}

40 u^{2} - 60 u - 40 = 0

Solve with the quadratic formula

40 u^{2} - 60 u - 40 = 0

Quadratic Equation Formula:

For

a = 40, b = - 60, c = - 40

u_{1, 2} = \frac{- ( - 60 ) \pm ( - 60 ) ^{2} - 4 \cdot 40 ( - 40 )}{2 \cdot 40}

u_{1, 2} = \frac{- ( - 60 ) \pm ( - 60 ) ^{2} - 4 \cdot 40 ( - 40 )}{2 \cdot 40}

(- 60)^{2} - 4 \cdot 40 (- 40) = 100

(- 60)^{2} - 4 \cdot 40 (- 40)

Apply rule

- (- a) = a

= (- 60)^{2} + 4 \cdot 40 \cdot 40

Apply exponent rule:

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

n

is even

(- 60)^{2} = 6 0^{2}

= 6 0^{2} + 4 \cdot 40 \cdot 40

Multiply the numbers:

4 \cdot 40 \cdot 40 = 6400

= 6 0^{2} + 6400

6 0^{2} = 3600

= 3600 + 6400

Add the numbers:

3600 + 6400 = 10000

= 10000

Factor the number:

10000 = 10 0^{2}

= 10 0^{2}

Apply radical rule:

10 0^{2} = 100

= 100

u_{1, 2} = \frac{- ( - 60 ) \pm 100}{2 \cdot 40}

Separate the solutions

u_{1} = \frac{- ( - 60 ) + 100}{2 \cdot 40}, u_{2} = \frac{- ( - 60 ) - 100}{2 \cdot 40}

u = \frac{- ( - 60 ) + 100}{2 \cdot 40} : 2

\frac{- ( - 60 ) + 100}{2 \cdot 40}

Apply rule

- (- a) = a

= \frac{60 + 100}{2 \cdot 40}

Add the numbers:

60 + 100 = 160

= \frac{160}{2 \cdot 40}

Multiply the numbers:

2 \cdot 40 = 80

= \frac{160}{80}

Divide the numbers:

\frac{160}{80} = 2

= 2

u = \frac{- ( - 60 ) - 100}{2 \cdot 40} : - \frac{1}{2}

\frac{- ( - 60 ) - 100}{2 \cdot 40}

Apply rule

- (- a) = a

= \frac{60 - 100}{2 \cdot 40}

Subtract the numbers:

60 - 100 = - 40

= \frac{- 40}{2 \cdot 40}

Multiply the numbers:

2 \cdot 40 = 80

= \frac{- 40}{80}

Apply the fraction rule:

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

= - \frac{40}{80}

Cancel the common factor:

40

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = 2, u = - \frac{1}{2}

u = 2, u = - \frac{1}{2}

Substitute back

u = v^{5},

solve for

v

Solve

v^{5} = 2

For

x^{n} = f (a)

, n is odd, the solution is

Solve

v^{5} = - \frac{1}{2}

For

x^{n} = f (a)

, n is odd, the solution is

Apply radical rule:

n

is odd

Apply radical rule:

The solutions are

Substitute back

v = u^{2},

solve for

u

Solve

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

Solve

No Solution for

u \in R

x^{2}

cannot be negative for

x \in R

No Solution for u \in R

The solutions are

Substitute back

u = v^{2},

solve for

v

Solve

Simplify

Apply radical rule:

= (2^{\frac{1}{5}})^{\frac{1}{2}}

Apply exponent rule:

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

= 2^{\frac{1}{5} \cdot \frac{1}{2}}

\frac{1}{5} \cdot \frac{1}{2} = \frac{1}{10}

\frac{1}{5} \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 1}{5 \cdot 2}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{5 \cdot 2}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{1}{10}

= 2^{\frac{1}{10}}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

Solve

No Solution for

v \in R

Simplify

Apply radical rule:

= (2^{\frac{1}{5}})^{\frac{1}{2}}

Apply exponent rule:

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

= 2^{\frac{1}{5} \cdot \frac{1}{2}}

\frac{1}{5} \cdot \frac{1}{2} = \frac{1}{10}

\frac{1}{5} \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 1}{5 \cdot 2}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{5 \cdot 2}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{1}{10}

= 2^{\frac{1}{10}}

x^{2}

cannot be negative for

x \in R

No Solution for v \in R

The solutions are

Substitute back

v = u^{2},

solve for

u

Solve

Simplify

Apply radical rule:

= (2^{\frac{1}{10}})^{\frac{1}{2}}

Apply exponent rule:

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

= 2^{\frac{1}{10} \cdot \frac{1}{2}}

\frac{1}{10} \cdot \frac{1}{2} = \frac{1}{20}

\frac{1}{10} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

10 \cdot 2 = 20

= \frac{1}{20}

= 2^{\frac{1}{20}}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

Solve

No Solution for

u \in R

Simplify

Apply radical rule:

= (2^{\frac{1}{10}})^{\frac{1}{2}}

Apply exponent rule:

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

= 2^{\frac{1}{10} \cdot \frac{1}{2}}

\frac{1}{10} \cdot \frac{1}{2} = \frac{1}{20}

\frac{1}{10} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

10 \cdot 2 = 20

= \frac{1}{20}

= 2^{\frac{1}{20}}

x^{2}

cannot be negative for

x \in R

No Solution for u \in R

The solutions are

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u^{40} - u^{- 40}) 40

and compare to zero

Solve

u^{40} = 0 : u = 0

u^{40} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

Substitute back

u = e^{x},

solve for

x

Solve

Apply exponent rules

Apply exponent rule:

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

Apply exponent rule:

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

e^{x} = 2^{\frac{1}{20} \cdot \frac{1}{2}}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (2^{\frac{1}{20} \cdot \frac{1}{2}})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (2^{\frac{1}{20} \cdot \frac{1}{2}})

Apply log rule:

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

ln (2^{\frac{1}{20} \cdot \frac{1}{2}}) = \frac{1}{20} \cdot \frac{1}{2} ln (2)

x = \frac{1}{20} \cdot \frac{1}{2} ln (2)

Simplify

x = \frac{1}{40} ln (2)

x = \frac{1}{40} ln (2)

Solve

No Solution for

x \in R

Apply exponent rules

Apply exponent rule:

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

e^{x} = - 2^{\frac{1}{20} \cdot \frac{1}{2}}

e^{x} = - 2^{\frac{1}{20} \cdot \frac{1}{2}}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = \frac{1}{40} ln (2)

x = \frac{1}{40} ln (2)

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x40)= 30/40 ?
The general solution for sinh(x40)= 30/40 is x= 1/40 ln(2)