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

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

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Popular Trigonometry >

sinh(x)= 4/3

Solution

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

Solution

x = ln (3)

Degrees

x = 62.94584 \dots^{\circ}

Solution steps

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

Rewrite using trig identities

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

Use the Hyperbolic identity:

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

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

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

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

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

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

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

Simplify

(e^{x} - e^{- x}) \cdot 3 = 8

Apply exponent rules

(e^{x} - e^{- x}) \cdot 3 = 8

Apply exponent rule:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 3 = 8

(e^{x} - (e^{x})^{- 1}) \cdot 3 = 8

Rewrite the equation with

e^{x} = u

(u - (u)^{- 1}) \cdot 3 = 8

Solve

(u - u^{- 1}) \cdot 3 = 8 : u = 3, u = - \frac{1}{3}

(u - u^{- 1}) \cdot 3 = 8

Refine

(u - \frac{1}{u}) \cdot 3 = 8

Simplify

(u - \frac{1}{u}) \cdot 3 : 3 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 3

Apply the commutative law:

(u - \frac{1}{u}) \cdot 3 = 3 (u - \frac{1}{u})

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

3 (u - \frac{1}{u}) = 8

Expand

3 (u - \frac{1}{u}) : 3 u - \frac{3}{u}

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

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = u, c = \frac{1}{u}

= 3 u - 3 \cdot \frac{1}{u}

3 \cdot \frac{1}{u} = \frac{3}{u}

3 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 3}{u}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{u}

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

3 u - \frac{3}{u} = 8

Multiply both sides by

u

3 u - \frac{3}{u} = 8

Multiply both sides by

u

3 uu - \frac{3}{u} u = 8 u

Simplify

3 uu - \frac{3}{u} u = 8 u

Simplify

3 uu : 3 u^{2}

3 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 3 u^{2}

Simplify

- \frac{3}{u} u : - 3

- \frac{3}{u} u

Multiply fractions:

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

= - \frac{3 u}{u}

Cancel the common factor:

u

= - 3

3 u^{2} - 3 = 8 u

3 u^{2} - 3 = 8 u

3 u^{2} - 3 = 8 u

Solve

3 u^{2} - 3 = 8 u : u = 3, u = - \frac{1}{3}

3 u^{2} - 3 = 8 u

Move

8 u

to the left side

3 u^{2} - 3 = 8 u

Subtract

8 u

from both sides

3 u^{2} - 3 - 8 u = 8 u - 8 u

Simplify

3 u^{2} - 3 - 8 u = 0

3 u^{2} - 3 - 8 u = 0

Write in the standard form

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

3 u^{2} - 8 u - 3 = 0

Solve with the quadratic formula

3 u^{2} - 8 u - 3 = 0

Quadratic Equation Formula:

For

a = 3, b = - 8, c = - 3

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 ( - 3 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 ( - 3 )}{2 \cdot 3}

(- 8)^{2} - 4 \cdot 3 (- 3) = 10

(- 8)^{2} - 4 \cdot 3 (- 3)

Apply rule

- (- a) = a

= (- 8)^{2} + 4 \cdot 3 \cdot 3

Apply exponent rule:

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

n

is even

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

= 8^{2} + 4 \cdot 3 \cdot 3

Multiply the numbers:

4 \cdot 3 \cdot 3 = 36

= 8^{2} + 36

8^{2} = 64

= 64 + 36

Add the numbers:

64 + 36 = 100

= 100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- ( - 8 ) \pm 10}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 10}{2 \cdot 3}, u_{2} = \frac{- ( - 8 ) - 10}{2 \cdot 3}

u = \frac{- ( - 8 ) + 10}{2 \cdot 3} : 3

\frac{- ( - 8 ) + 10}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{8 + 10}{2 \cdot 3}

Add the numbers:

8 + 10 = 18

= \frac{18}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{18}{6}

Divide the numbers:

\frac{18}{6} = 3

= 3

u = \frac{- ( - 8 ) - 10}{2 \cdot 3} : - \frac{1}{3}

\frac{- ( - 8 ) - 10}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{8 - 10}{2 \cdot 3}

Subtract the numbers:

8 - 10 = - 2

= \frac{- 2}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 2}{6}

Apply the fraction rule:

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

= - \frac{2}{6}

Cancel the common factor:

2

= - \frac{1}{3}

The solutions to the quadratic equation are:

u = 3, u = - \frac{1}{3}

u = 3, u = - \frac{1}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u - u^{- 1}) 3

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 3, u = - \frac{1}{3}

u = 3, u = - \frac{1}{3}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 3 : x = ln (3)

e^{x} = 3

Apply exponent rules

e^{x} = 3

f (x) = g (x)

, then

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

ln (e^{x}) = ln (3)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3)

x = ln (3)

Solve

e^{x} = - \frac{1}{3} :

No Solution for

x \in R

e^{x} = - \frac{1}{3}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (3)

x = ln (3)

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