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

The general solution for sinh(4x)= 3/4 is x= 1/4 ln(2)

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sinh(4x)= 3/4

Solution

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

Solution

x = \frac{1}{4} ln (2)

Degrees

x = 9.92860 \dots^{\circ}

Solution steps

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

Rewrite using trig identities

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

Use the Hyperbolic identity:

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

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

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

\frac{e ^{4 x} - e ^{- 4 x}}{2} = \frac{3}{4} : x = \frac{1}{4} ln (2)

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

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

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

Simplify

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

Apply exponent rules

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

Apply exponent rule:

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

e^{4 x} = (e^{x})^{4}, e^{- 4 x} = (e^{x})^{- 4}

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

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

Rewrite the equation with

e^{x} = u

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

Solve

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

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

Refine

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

Simplify

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

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

Apply the commutative law:

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{u ^{4}}

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

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

Multiply both sides by

u^{4}

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

Multiply both sides by

u^{4}

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

Simplify

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

Simplify

4 u^{4} u^{4} : 4 u^{8}

4 u^{4} u^{4}

Apply exponent rule:

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

u^{4} u^{4} = u^{4 + 4}

= 4 u^{4 + 4}

Add the numbers:

4 + 4 = 8

= 4 u^{8}

Simplify

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

- \frac{4}{u ^{4}} u^{4}

Multiply fractions:

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

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

Cancel the common factor:

u^{4}

= - 4

4 u^{8} - 4 = 6 u^{4}

4 u^{8} - 4 = 6 u^{4}

4 u^{8} - 4 = 6 u^{4}

Solve

4 u^{8} - 4 = 6 u^{4} : u = 2, u = - 2

4 u^{8} - 4 = 6 u^{4}

Move

6 u^{4}

to the left side

4 u^{8} - 4 = 6 u^{4}

Subtract

6 u^{4}

from both sides

4 u^{8} - 4 - 6 u^{4} = 6 u^{4} - 6 u^{4}

Simplify

4 u^{8} - 4 - 6 u^{4} = 0

4 u^{8} - 4 - 6 u^{4} = 0

Write in the standard form

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

4 u^{8} - 6 u^{4} - 4 = 0

Rewrite the equation with

v = u^{2}, v^{2} = u^{4}

and

v^{4} = u^{8}

4 v^{4} - 6 v^{2} - 4 = 0

Solve

4 v^{4} - 6 v^{2} - 4 = 0 : v = 2, v = - 2

4 v^{4} - 6 v^{2} - 4 = 0

Rewrite the equation with

u = v^{2}

and

u^{2} = v^{4}

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

Solve

4 u^{2} - 6 u - 4 = 0 : u = 2, u = - \frac{1}{2}

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

Apply exponent rule:

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

n

is even

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

= 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} : 2

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

Apply rule

- (- a) = a

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

Add the numbers:

6 + 10 = 16

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{16}{8}

Divide the numbers:

\frac{16}{8} = 2

= 2

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

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

Apply rule

- (- a) = a

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

Subtract the numbers:

6 - 10 = - 4

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4}{8}

Apply the fraction rule:

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

= - \frac{4}{8}

Cancel the common factor:

4

= - \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^{2},

solve for

v

Solve

v^{2} = 2 : v = 2, v = - 2

v^{2} = 2

For

x^{2} = f (a)

the solutions are

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

v = 2, v = - 2

Solve

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

No Solution for

v \in R

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

x^{2}

cannot be negative for

x \in R

No Solution for v \in R

The solutions are

v = 2, v = - 2

v = 2, v = - 2

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 2 : u = 2, u = - 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

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

u = 2, u = - 2

Solve

u^{2} = - 2 :

No Solution for

u \in R

u^{2} = - 2

x^{2}

cannot be negative for

x \in R

No Solution for u \in R

The solutions are

u = 2, u = - 2

u = 2, u = - 2

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u^{4} - u^{- 4}) 4

and compare to zero

Solve

u^{4} = 0 : u = 0

u^{4} = 0

Apply rule

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

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2, u = - 2

u = 2, u = - 2

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 2 : x = \frac{1}{4} ln (2)

e^{x} = 2

Apply exponent rules

e^{x} = 2

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

f (x) = g (x)

, then

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

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

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

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

Apply log rule:

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

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

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

Simplify

x = \frac{1}{4} ln (2)

x = \frac{1}{4} ln (2)

Solve

e^{x} = - 2 :

No Solution for

x \in R

e^{x} = - 2

Apply exponent rules

e^{x} = - 2

Apply exponent rule:

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

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

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

e^{x} = - 2^{\frac{1}{2} \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}{4} ln (2)

x = \frac{1}{4} ln (2)

Popular Examples

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tan (x) + sec (x) = 3

cos^2(3x+pi/4)= 3/4

cos^{2} (3 x + \frac{π}{4}) = \frac{3}{4}

2sin(pi/3-x)-1=0

2 sin (\frac{π}{3} - x) - 1 = 0

-sqrt(2)sin(x)-1=0

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(4x)= 3/4 ?
The general solution for sinh(4x)= 3/4 is x= 1/4 ln(2)