What is the general solution for 3sinh(x)-cosh(x)=1 ?

The general solution for 3sinh(x)-cosh(x)=1 is x=ln(2)

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Popular Trigonometry >

3sinh(x)-cosh(x)=1

Solution

3 sinh (x) - cosh (x) = 1

Solution

x = ln (2)

Degrees

x = 39.71440 \dots^{\circ}

Solution steps

3 sinh (x) - cosh (x) = 1

Rewrite using trig identities

3 sinh (x) - cosh (x) = 1

Use the Hyperbolic identity:

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

3 \cdot \frac{e ^{x} - e ^{- x}}{2} - cosh (x) = 1

Use the Hyperbolic identity:

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

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

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

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

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

Multiply both sides by

2

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

Simplify

3 (e^{x} - e^{- x}) - (e^{x} + e^{- x}) = 2

Apply exponent rules

3 (e^{x} - e^{- x}) - (e^{x} + e^{- x}) = 2

Apply exponent rule:

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

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

3 (e^{x} - (e^{x})^{- 1}) - (e^{x} + (e^{x})^{- 1}) = 2

3 (e^{x} - (e^{x})^{- 1}) - (e^{x} + (e^{x})^{- 1}) = 2

Rewrite the equation with

e^{x} = u

3 (u - (u)^{- 1}) - (u + (u)^{- 1}) = 2

Solve

3 (u - u^{- 1}) - (u + u^{- 1}) = 2 : u = 2, u = - 1

3 (u - u^{- 1}) - (u + u^{- 1}) = 2

Refine

3 (u - \frac{1}{u}) - (u + \frac{1}{u}) = 2

Simplify

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

3 (u - \frac{1}{u}) - u - \frac{1}{u} = 2

Multiply both sides by

u

3 (u - \frac{1}{u}) - u - \frac{1}{u} = 2

Multiply both sides by

u

3 (u - \frac{1}{u}) u - uu - \frac{1}{u} u = 2 u

Simplify

3 (u - \frac{1}{u}) u - uu - \frac{1}{u} u = 2 u

Simplify

- uu : - u^{2}

- uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - u^{2}

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

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

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

Cancel the common factor:

u

= - 1

3 (u - \frac{1}{u}) u - u^{2} - 1 = 2 u

3 (u - \frac{1}{u}) u - u^{2} - 1 = 2 u

3 (u - \frac{1}{u}) u - u^{2} - 1 = 2 u

Expand

3 (u - \frac{1}{u}) u - u^{2} - 1 : 2 u^{2} - 4

3 (u - \frac{1}{u}) u - u^{2} - 1

= 3 u (u - \frac{1}{u}) - u^{2} - 1

Expand

3 u (u - \frac{1}{u}) : 3 u^{2} - 3

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

Simplify

3 uu - 3 \cdot \frac{1}{u} u : 3 u^{2} - 3

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

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}

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

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

Multiply fractions:

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

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

Cancel the common factor:

u

= 1 \cdot 3

Multiply the numbers:

1 \cdot 3 = 3

= 3

= 3 u^{2} - 3

= 3 u^{2} - 3

= 3 u^{2} - 3 - u^{2} - 1

Simplify

3 u^{2} - 3 - u^{2} - 1 : 2 u^{2} - 4

3 u^{2} - 3 - u^{2} - 1

Group like terms

= 3 u^{2} - u^{2} - 3 - 1

Add similar elements:

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

= 2 u^{2} - 3 - 1

Subtract the numbers:

- 3 - 1 = - 4

= 2 u^{2} - 4

= 2 u^{2} - 4

2 u^{2} - 4 = 2 u

Solve

2 u^{2} - 4 = 2 u : u = 2, u = - 1

2 u^{2} - 4 = 2 u

Move

2 u

to the left side

2 u^{2} - 4 = 2 u

Subtract

2 u

from both sides

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

Simplify

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 2, b = - 2, c = - 4

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

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

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

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 2 \cdot 4 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

Add the numbers:

4 + 32 = 36

= 36

Factor the number:

36 = 6^{2}

= 6^{2}

Apply radical rule:

6^{2} = 6

= 6

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

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 6}{2 \cdot 2}, u_{2} = \frac{- ( - 2 ) - 6}{2 \cdot 2}

u = \frac{- ( - 2 ) + 6}{2 \cdot 2} : 2

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

Apply rule

- (- a) = a

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

Add the numbers:

2 + 6 = 8

= \frac{8}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

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

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

Apply rule

- (- a) = a

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

Subtract the numbers:

2 - 6 = - 4

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 4}{4}

Apply the fraction rule:

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

= - \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = 2, u = - 1

u = 2, u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

3 (u - u^{- 1}) - (u + u^{- 1})

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2, u = - 1

u = 2, u = - 1

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 2 : x = ln (2)

e^{x} = 2

Apply exponent rules

e^{x} = 2

f (x) = g (x)

, then

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

ln (e^{x}) = ln (2)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (2)

x = ln (2)

Solve

e^{x} = - 1 :

No Solution for

x \in R

e^{x} = - 1

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (2)

x = ln (2)

Graph

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Frequently Asked Questions (FAQ)

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