What is the general solution for sinh(x)= 36/77 ?

The general solution for sinh(x)= 36/77 is x=ln(11/7)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

sinh(x)= 36/77

Solution

sinh (x) = \frac{36}{77}

Solution

x = ln (\frac{11}{7})

Degrees

x = 25.89683 \dots^{\circ}

Solution steps

sinh (x) = \frac{36}{77}

Rewrite using trig identities

sinh (x) = \frac{36}{77}

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{36}{77}

\frac{e ^{x} - e ^{- x}}{2} = \frac{36}{77}

\frac{e ^{x} - e ^{- x}}{2} = \frac{36}{77} : x = ln (\frac{11}{7})

\frac{e ^{x} - e ^{- x}}{2} = \frac{36}{77}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 77 = 2 \cdot 36

Simplify

(e^{x} - e^{- x}) \cdot 77 = 72

Apply exponent rules

(e^{x} - e^{- x}) \cdot 77 = 72

Apply exponent rule:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 77 = 72

(e^{x} - (e^{x})^{- 1}) \cdot 77 = 72

Rewrite the equation with

e^{x} = u

(u - (u)^{- 1}) \cdot 77 = 72

Solve

(u - u^{- 1}) \cdot 77 = 72 : u = \frac{11}{7}, u = - \frac{7}{11}

(u - u^{- 1}) \cdot 77 = 72

Refine

(u - \frac{1}{u}) \cdot 77 = 72

Simplify

(u - \frac{1}{u}) \cdot 77 : 77 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 77

Apply the commutative law:

(u - \frac{1}{u}) \cdot 77 = 77 (u - \frac{1}{u})

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

77 (u - \frac{1}{u}) = 72

Expand

77 (u - \frac{1}{u}) : 77 u - \frac{77}{u}

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

Apply the distributive law:

a (b - c) = ab - a c

a = 77, b = u, c = \frac{1}{u}

= 77 u - 77 \cdot \frac{1}{u}

77 \cdot \frac{1}{u} = \frac{77}{u}

77 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 77}{u}

Multiply the numbers:

1 \cdot 77 = 77

= \frac{77}{u}

= 77 u - \frac{77}{u}

77 u - \frac{77}{u} = 72

Multiply both sides by

u

77 u - \frac{77}{u} = 72

Multiply both sides by

u

77 uu - \frac{77}{u} u = 72 u

Simplify

77 uu - \frac{77}{u} u = 72 u

Simplify

77 uu : 77 u^{2}

77 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 77 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 77 u^{2}

Simplify

- \frac{77}{u} u : - 77

- \frac{77}{u} u

Multiply fractions:

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

= - \frac{77 u}{u}

Cancel the common factor:

u

= - 77

77 u^{2} - 77 = 72 u

77 u^{2} - 77 = 72 u

77 u^{2} - 77 = 72 u

Solve

77 u^{2} - 77 = 72 u : u = \frac{11}{7}, u = - \frac{7}{11}

77 u^{2} - 77 = 72 u

Move

72 u

to the left side

77 u^{2} - 77 = 72 u

Subtract

72 u

from both sides

77 u^{2} - 77 - 72 u = 72 u - 72 u

Simplify

77 u^{2} - 77 - 72 u = 0

77 u^{2} - 77 - 72 u = 0

Write in the standard form

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

77 u^{2} - 72 u - 77 = 0

Solve with the quadratic formula

77 u^{2} - 72 u - 77 = 0

Quadratic Equation Formula:

For

a = 77, b = - 72, c = - 77

u_{1, 2} = \frac{- ( - 72 ) \pm ( - 72 ) ^{2} - 4 \cdot 77 ( - 77 )}{2 \cdot 77}

u_{1, 2} = \frac{- ( - 72 ) \pm ( - 72 ) ^{2} - 4 \cdot 77 ( - 77 )}{2 \cdot 77}

(- 72)^{2} - 4 \cdot 77 (- 77) = 170

(- 72)^{2} - 4 \cdot 77 (- 77)

Apply rule

- (- a) = a

= (- 72)^{2} + 4 \cdot 77 \cdot 77

Apply exponent rule:

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

n

is even

(- 72)^{2} = 7 2^{2}

= 7 2^{2} + 4 \cdot 77 \cdot 77

Multiply the numbers:

4 \cdot 77 \cdot 77 = 23716

= 7 2^{2} + 23716

7 2^{2} = 5184

= 5184 + 23716

Add the numbers:

5184 + 23716 = 28900

= 28900

Factor the number:

28900 = 17 0^{2}

= 17 0^{2}

Apply radical rule:

17 0^{2} = 170

= 170

u_{1, 2} = \frac{- ( - 72 ) \pm 170}{2 \cdot 77}

Separate the solutions

u_{1} = \frac{- ( - 72 ) + 170}{2 \cdot 77}, u_{2} = \frac{- ( - 72 ) - 170}{2 \cdot 77}

u = \frac{- ( - 72 ) + 170}{2 \cdot 77} : \frac{11}{7}

\frac{- ( - 72 ) + 170}{2 \cdot 77}

Apply rule

- (- a) = a

= \frac{72 + 170}{2 \cdot 77}

Add the numbers:

72 + 170 = 242

= \frac{242}{2 \cdot 77}

Multiply the numbers:

2 \cdot 77 = 154

= \frac{242}{154}

Cancel the common factor:

22

= \frac{11}{7}

u = \frac{- ( - 72 ) - 170}{2 \cdot 77} : - \frac{7}{11}

\frac{- ( - 72 ) - 170}{2 \cdot 77}

Apply rule

- (- a) = a

= \frac{72 - 170}{2 \cdot 77}

Subtract the numbers:

72 - 170 = - 98

= \frac{- 98}{2 \cdot 77}

Multiply the numbers:

2 \cdot 77 = 154

= \frac{- 98}{154}

Apply the fraction rule:

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

= - \frac{98}{154}

Cancel the common factor:

14

= - \frac{7}{11}

The solutions to the quadratic equation are:

u = \frac{11}{7}, u = - \frac{7}{11}

u = \frac{11}{7}, u = - \frac{7}{11}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u - u^{- 1}) 77

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{11}{7}, u = - \frac{7}{11}

u = \frac{11}{7}, u = - \frac{7}{11}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{11}{7} : x = ln (\frac{11}{7})

e^{x} = \frac{11}{7}

Apply exponent rules

e^{x} = \frac{11}{7}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{11}{7})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{11}{7})

x = ln (\frac{11}{7})

Solve

e^{x} = - \frac{7}{11} :

No Solution for

x \in R

e^{x} = - \frac{7}{11}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (\frac{11}{7})

x = ln (\frac{11}{7})

Graph

Popular Examples

16cos(θ)=4

-3sin(2x)=5cos(2x)

0.001=(sin(θ)*12^2)/(9.81)

2-5cos(x)=0

4cos(2x)-2=0

Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)= 36/77 ?
The general solution for sinh(x)= 36/77 is x=ln(11/7)