What is the general solution for sinh(x)= 33/56 ?

The general solution for sinh(x)= 33/56 is x=ln(7/4)

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Popular Trigonometry >

sinh(x)= 33/56

Solution

sinh (x) = \frac{33}{56}

Solution

x = ln (\frac{7}{4})

Degrees

x = 32.06362 \dots^{\circ}

Solution steps

sinh (x) = \frac{33}{56}

Rewrite using trig identities

sinh (x) = \frac{33}{56}

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{33}{56}

\frac{e ^{x} - e ^{- x}}{2} = \frac{33}{56}

\frac{e ^{x} - e ^{- x}}{2} = \frac{33}{56} : x = ln (\frac{7}{4})

\frac{e ^{x} - e ^{- x}}{2} = \frac{33}{56}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 56 = 2 \cdot 33

Simplify

(e^{x} - e^{- x}) \cdot 56 = 66

Apply exponent rules

(e^{x} - e^{- x}) \cdot 56 = 66

Apply exponent rule:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 56 = 66

(e^{x} - (e^{x})^{- 1}) \cdot 56 = 66

Rewrite the equation with

e^{x} = u

(u - (u)^{- 1}) \cdot 56 = 66

Solve

(u - u^{- 1}) \cdot 56 = 66 : u = \frac{7}{4}, u = - \frac{4}{7}

(u - u^{- 1}) \cdot 56 = 66

Refine

(u - \frac{1}{u}) \cdot 56 = 66

Simplify

(u - \frac{1}{u}) \cdot 56 : 56 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 56

Apply the commutative law:

(u - \frac{1}{u}) \cdot 56 = 56 (u - \frac{1}{u})

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

56 (u - \frac{1}{u}) = 66

Expand

56 (u - \frac{1}{u}) : 56 u - \frac{56}{u}

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

Apply the distributive law:

a (b - c) = ab - a c

a = 56, b = u, c = \frac{1}{u}

= 56 u - 56 \cdot \frac{1}{u}

56 \cdot \frac{1}{u} = \frac{56}{u}

56 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 56}{u}

Multiply the numbers:

1 \cdot 56 = 56

= \frac{56}{u}

= 56 u - \frac{56}{u}

56 u - \frac{56}{u} = 66

Multiply both sides by

u

56 u - \frac{56}{u} = 66

Multiply both sides by

u

56 uu - \frac{56}{u} u = 66 u

Simplify

56 uu - \frac{56}{u} u = 66 u

Simplify

56 uu : 56 u^{2}

56 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 56 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 56 u^{2}

Simplify

- \frac{56}{u} u : - 56

- \frac{56}{u} u

Multiply fractions:

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

= - \frac{56 u}{u}

Cancel the common factor:

u

= - 56

56 u^{2} - 56 = 66 u

56 u^{2} - 56 = 66 u

56 u^{2} - 56 = 66 u

Solve

56 u^{2} - 56 = 66 u : u = \frac{7}{4}, u = - \frac{4}{7}

56 u^{2} - 56 = 66 u

Move

66 u

to the left side

56 u^{2} - 56 = 66 u

Subtract

66 u

from both sides

56 u^{2} - 56 - 66 u = 66 u - 66 u

Simplify

56 u^{2} - 56 - 66 u = 0

56 u^{2} - 56 - 66 u = 0

Write in the standard form

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

56 u^{2} - 66 u - 56 = 0

Solve with the quadratic formula

56 u^{2} - 66 u - 56 = 0

Quadratic Equation Formula:

For

a = 56, b = - 66, c = - 56

u_{1, 2} = \frac{- ( - 66 ) \pm ( - 66 ) ^{2} - 4 \cdot 56 ( - 56 )}{2 \cdot 56}

u_{1, 2} = \frac{- ( - 66 ) \pm ( - 66 ) ^{2} - 4 \cdot 56 ( - 56 )}{2 \cdot 56}

(- 66)^{2} - 4 \cdot 56 (- 56) = 130

(- 66)^{2} - 4 \cdot 56 (- 56)

Apply rule

- (- a) = a

= (- 66)^{2} + 4 \cdot 56 \cdot 56

Apply exponent rule:

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

n

is even

(- 66)^{2} = 6 6^{2}

= 6 6^{2} + 4 \cdot 56 \cdot 56

Multiply the numbers:

4 \cdot 56 \cdot 56 = 12544

= 6 6^{2} + 12544

6 6^{2} = 4356

= 4356 + 12544

Add the numbers:

4356 + 12544 = 16900

= 16900

Factor the number:

16900 = 13 0^{2}

= 13 0^{2}

Apply radical rule:

n a^{n} = a

13 0^{2} = 130

= 130

u_{1, 2} = \frac{- ( - 66 ) \pm 130}{2 \cdot 56}

Separate the solutions

u_{1} = \frac{- ( - 66 ) + 130}{2 \cdot 56}, u_{2} = \frac{- ( - 66 ) - 130}{2 \cdot 56}

u = \frac{- ( - 66 ) + 130}{2 \cdot 56} : \frac{7}{4}

\frac{- ( - 66 ) + 130}{2 \cdot 56}

Apply rule

- (- a) = a

= \frac{66 + 130}{2 \cdot 56}

Add the numbers:

66 + 130 = 196

= \frac{196}{2 \cdot 56}

Multiply the numbers:

2 \cdot 56 = 112

= \frac{196}{112}

Cancel the common factor:

28

= \frac{7}{4}

u = \frac{- ( - 66 ) - 130}{2 \cdot 56} : - \frac{4}{7}

\frac{- ( - 66 ) - 130}{2 \cdot 56}

Apply rule

- (- a) = a

= \frac{66 - 130}{2 \cdot 56}

Subtract the numbers:

66 - 130 = - 64

= \frac{- 64}{2 \cdot 56}

Multiply the numbers:

2 \cdot 56 = 112

= \frac{- 64}{112}

Apply the fraction rule:

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

= - \frac{64}{112}

Cancel the common factor:

16

= - \frac{4}{7}

The solutions to the quadratic equation are:

u = \frac{7}{4}, u = - \frac{4}{7}

u = \frac{7}{4}, u = - \frac{4}{7}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u - u^{- 1}) 56

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{7}{4}, u = - \frac{4}{7}

u = \frac{7}{4}, u = - \frac{4}{7}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{7}{4} : x = ln (\frac{7}{4})

e^{x} = \frac{7}{4}

Apply exponent rules

e^{x} = \frac{7}{4}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{7}{4})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{7}{4})

x = ln (\frac{7}{4})

Solve

e^{x} = - \frac{4}{7} :

No Solution for

x \in R

e^{x} = - \frac{4}{7}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (\frac{7}{4})

x = ln (\frac{7}{4})

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)= 33/56 ?
The general solution for sinh(x)= 33/56 is x=ln(7/4)