What is the general solution for sinh(x)=(-3)/4 ?

The general solution for sinh(x)=(-3)/4 is x=-ln(2)

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sinh(x)=(-3)/4

Solution

sinh (x) = \frac{- 3}{4}

Solution

x = - ln (2)

Degrees

x = - 39.71440 \dots^{\circ}

Solution steps

sinh (x) = \frac{- 3}{4}

Rewrite using trig identities

sinh (x) = \frac{- 3}{4}

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{- 3}{4}

\frac{e ^{x} - e ^{- x}}{2} = \frac{- 3}{4}

\frac{e ^{x} - e ^{- x}}{2} = \frac{- 3}{4} : x = - ln (2)

\frac{e ^{x} - e ^{- x}}{2} = \frac{- 3}{4}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 4 = 2 (- 3)

Simplify

(e^{x} - e^{- x}) \cdot 4 = - 6

Apply exponent rules

(e^{x} - e^{- x}) \cdot 4 = - 6

Apply exponent rule:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 4 = - 6

(e^{x} - (e^{x})^{- 1}) \cdot 4 = - 6

Rewrite the equation with

e^{x} = u

(u - (u)^{- 1}) \cdot 4 = - 6

Solve

(u - u^{- 1}) \cdot 4 = - 6 : u = \frac{1}{2}, u = - 2

(u - u^{- 1}) \cdot 4 = - 6

Refine

(u - \frac{1}{u}) \cdot 4 = - 6

Simplify

(u - \frac{1}{u}) \cdot 4 : 4 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 4

Apply the commutative law:

(u - \frac{1}{u}) \cdot 4 = 4 (u - \frac{1}{u})

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

4 (u - \frac{1}{u}) = - 6

Expand

4 (u - \frac{1}{u}) : 4 u - \frac{4}{u}

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

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = u, c = \frac{1}{u}

= 4 u - 4 \cdot \frac{1}{u}

4 \cdot \frac{1}{u} = \frac{4}{u}

4 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 4}{u}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{u}

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

4 u - \frac{4}{u} = - 6

Multiply both sides by

u

4 u - \frac{4}{u} = - 6

Multiply both sides by

u

4 uu - \frac{4}{u} u = - 6 u

Simplify

4 uu - \frac{4}{u} u = - 6 u

Simplify

4 uu : 4 u^{2}

4 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 4 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 4 u^{2}

Simplify

- \frac{4}{u} u : - 4

- \frac{4}{u} u

Multiply fractions:

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

= - \frac{4 u}{u}

Cancel the common factor:

u

= - 4

4 u^{2} - 4 = - 6 u

4 u^{2} - 4 = - 6 u

4 u^{2} - 4 = - 6 u

Solve

4 u^{2} - 4 = - 6 u : u = \frac{1}{2}, u = - 2

4 u^{2} - 4 = - 6 u

Move

6 u

to the left side

4 u^{2} - 4 = - 6 u

Add

6 u

to both sides

4 u^{2} - 4 + 6 u = - 6 u + 6 u

Simplify

4 u^{2} - 4 + 6 u = 0

4 u^{2} - 4 + 6 u = 0

Write in the standard form

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

4 u^{2} + 6 u - 4 = 0

Solve with the quadratic formula

4 u^{2} + 6 u - 4 = 0

Quadratic Equation Formula:

For

a = 4, b = 6, c = - 4

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 4 ( - 4 )}{2 \cdot 4}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 4 ( - 4 )}{2 \cdot 4}

6^{2} - 4 \cdot 4 (- 4) = 10

6^{2} - 4 \cdot 4 (- 4)

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 4 \cdot 4 = 64

= 6^{2} + 64

6^{2} = 36

= 36 + 64

Add the numbers:

36 + 64 = 100

= 100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 6 \pm 10}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 6 + 10}{2 \cdot 4}, u_{2} = \frac{- 6 - 10}{2 \cdot 4}

u = \frac{- 6 + 10}{2 \cdot 4} : \frac{1}{2}

\frac{- 6 + 10}{2 \cdot 4}

Add/Subtract the numbers:

- 6 + 10 = 4

= \frac{4}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

u = \frac{- 6 - 10}{2 \cdot 4} : - 2

\frac{- 6 - 10}{2 \cdot 4}

Subtract the numbers:

- 6 - 10 = - 16

= \frac{- 16}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 16}{8}

Apply the fraction rule:

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

= - \frac{16}{8}

Divide the numbers:

\frac{16}{8} = 2

= - 2

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u - u^{- 1}) 4

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

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

Substitute back

u = e^{x},

solve for

x

Solve

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

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

Apply exponent rules

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

f (x) = g (x)

, then

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

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

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

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

Simplify

ln (\frac{1}{2}) : - ln (2)

ln (\frac{1}{2})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (2)

x = - ln (2)

x = - ln (2)

Solve

e^{x} = - 2 :

No Solution for

x \in R

e^{x} = - 2

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = - ln (2)

x = - ln (2)

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)=(-3)/4 ?
The general solution for sinh(x)=(-3)/4 is x=-ln(2)