What is the general solution for sinh(x)= 9/40 ?

The general solution for sinh(x)= 9/40 is x=ln(5/4)

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Popular Trigonometry >

sinh(x)= 9/40

Solution

sinh (x) = \frac{9}{40}

Solution

x = ln (\frac{5}{4})

Degrees

x = 12.78518 \dots^{\circ}

Solution steps

sinh (x) = \frac{9}{40}

Rewrite using trig identities

sinh (x) = \frac{9}{40}

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{9}{40}

\frac{e ^{x} - e ^{- x}}{2} = \frac{9}{40}

\frac{e ^{x} - e ^{- x}}{2} = \frac{9}{40} : x = ln (\frac{5}{4})

\frac{e ^{x} - e ^{- x}}{2} = \frac{9}{40}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 40 = 2 \cdot 9

Simplify

(e^{x} - e^{- x}) \cdot 40 = 18

Apply exponent rules

(e^{x} - e^{- x}) \cdot 40 = 18

Apply exponent rule:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 40 = 18

(e^{x} - (e^{x})^{- 1}) \cdot 40 = 18

Rewrite the equation with

e^{x} = u

(u - (u)^{- 1}) \cdot 40 = 18

Solve

(u - u^{- 1}) \cdot 40 = 18 : u = \frac{5}{4}, u = - \frac{4}{5}

(u - u^{- 1}) \cdot 40 = 18

Refine

(u - \frac{1}{u}) \cdot 40 = 18

Simplify

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

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

Apply the commutative law:

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

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

40 (u - \frac{1}{u}) = 18

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

40 \cdot \frac{1}{u}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 40 = 40

= \frac{40}{u}

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

40 u - \frac{40}{u} = 18

Multiply both sides by

u

40 u - \frac{40}{u} = 18

Multiply both sides by

u

40 uu - \frac{40}{u} u = 18 u

Simplify

40 uu - \frac{40}{u} u = 18 u

Simplify

40 uu : 40 u^{2}

40 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 40 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 40 u^{2}

Simplify

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

- \frac{40}{u} u

Multiply fractions:

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

= - \frac{40 u}{u}

Cancel the common factor:

u

= - 40

40 u^{2} - 40 = 18 u

40 u^{2} - 40 = 18 u

40 u^{2} - 40 = 18 u

Solve

40 u^{2} - 40 = 18 u : u = \frac{5}{4}, u = - \frac{4}{5}

40 u^{2} - 40 = 18 u

Move

18 u

to the left side

40 u^{2} - 40 = 18 u

Subtract

18 u

from both sides

40 u^{2} - 40 - 18 u = 18 u - 18 u

Simplify

40 u^{2} - 40 - 18 u = 0

40 u^{2} - 40 - 18 u = 0

Write in the standard form

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

40 u^{2} - 18 u - 40 = 0

Solve with the quadratic formula

40 u^{2} - 18 u - 40 = 0

Quadratic Equation Formula:

For

a = 40, b = - 18, c = - 40

u_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 \cdot 40 ( - 40 )}{2 \cdot 40}

u_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 \cdot 40 ( - 40 )}{2 \cdot 40}

(- 18)^{2} - 4 \cdot 40 (- 40) = 82

(- 18)^{2} - 4 \cdot 40 (- 40)

Apply rule

- (- a) = a

= (- 18)^{2} + 4 \cdot 40 \cdot 40

Apply exponent rule:

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

n

is even

(- 18)^{2} = 1 8^{2}

= 1 8^{2} + 4 \cdot 40 \cdot 40

Multiply the numbers:

4 \cdot 40 \cdot 40 = 6400

= 1 8^{2} + 6400

1 8^{2} = 324

= 324 + 6400

Add the numbers:

324 + 6400 = 6724

= 6724

Factor the number:

6724 = 8 2^{2}

= 8 2^{2}

Apply radical rule:

8 2^{2} = 82

= 82

u_{1, 2} = \frac{- ( - 18 ) \pm 82}{2 \cdot 40}

Separate the solutions

u_{1} = \frac{- ( - 18 ) + 82}{2 \cdot 40}, u_{2} = \frac{- ( - 18 ) - 82}{2 \cdot 40}

u = \frac{- ( - 18 ) + 82}{2 \cdot 40} : \frac{5}{4}

\frac{- ( - 18 ) + 82}{2 \cdot 40}

Apply rule

- (- a) = a

= \frac{18 + 82}{2 \cdot 40}

Add the numbers:

18 + 82 = 100

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

Multiply the numbers:

2 \cdot 40 = 80

= \frac{100}{80}

Cancel the common factor:

20

= \frac{5}{4}

u = \frac{- ( - 18 ) - 82}{2 \cdot 40} : - \frac{4}{5}

\frac{- ( - 18 ) - 82}{2 \cdot 40}

Apply rule

- (- a) = a

= \frac{18 - 82}{2 \cdot 40}

Subtract the numbers:

18 - 82 = - 64

= \frac{- 64}{2 \cdot 40}

Multiply the numbers:

2 \cdot 40 = 80

= \frac{- 64}{80}

Apply the fraction rule:

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

= - \frac{64}{80}

Cancel the common factor:

16

= - \frac{4}{5}

The solutions to the quadratic equation are:

u = \frac{5}{4}, u = - \frac{4}{5}

u = \frac{5}{4}, u = - \frac{4}{5}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u - u^{- 1}) 40

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{5}{4}, u = - \frac{4}{5}

u = \frac{5}{4}, u = - \frac{4}{5}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{5}{4} : x = ln (\frac{5}{4})

e^{x} = \frac{5}{4}

Apply exponent rules

e^{x} = \frac{5}{4}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{5}{4})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{5}{4})

x = ln (\frac{5}{4})

Solve

e^{x} = - \frac{4}{5} :

No Solution for

x \in R

e^{x} = - \frac{4}{5}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (\frac{5}{4})

x = ln (\frac{5}{4})

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)= 9/40 ?
The general solution for sinh(x)= 9/40 is x=ln(5/4)