What is the general solution for sinh(z)-cosh(z)=0 ?

The general solution for sinh(z)-cosh(z)=0 is No Solution for z\in\mathbb{R}

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sinh(z)-cosh(z)=0

Solution

sinh (z) - cosh (z) = 0

Solution

No Solution for z \in R

Solution steps

sinh (z) - cosh (z) = 0

Rewrite using trig identities

sinh (z) - cosh (z) = 0

Use the Hyperbolic identity:

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

\frac{e ^{z} - e ^{- z}}{2} - cosh (z) = 0

Use the Hyperbolic identity:

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

\frac{e ^{z} - e ^{- z}}{2} - \frac{e ^{z} + e ^{- z}}{2} = 0

\frac{e ^{z} - e ^{- z}}{2} - \frac{e ^{z} + e ^{- z}}{2} = 0

\frac{e ^{z} - e ^{- z}}{2} - \frac{e ^{z} + e ^{- z}}{2} = 0 :

No Solution for

z \in R

\frac{e ^{z} - e ^{- z}}{2} - \frac{e ^{z} + e ^{- z}}{2} = 0

Multiply both sides by

2

\frac{e ^{z} - e ^{- z}}{2} \cdot 2 - \frac{e ^{z} + e ^{- z}}{2} \cdot 2 = 0 \cdot 2

Simplify

e^{z} - e^{- z} - (e^{z} + e^{- z}) = 0

Apply exponent rules

e^{z} - e^{- z} - (e^{z} + e^{- z}) = 0

Apply exponent rule:

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

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

e^{z} - (e^{z})^{- 1} - (e^{z} + (e^{z})^{- 1}) = 0

e^{z} - (e^{z})^{- 1} - (e^{z} + (e^{z})^{- 1}) = 0

Rewrite the equation with

e^{z} = u

u - (u)^{- 1} - (u + (u)^{- 1}) = 0

Solve

u - u^{- 1} - (u + u^{- 1}) = 0 :

No Solution for

u \in R

u - u^{- 1} - (u + u^{- 1}) = 0

Refine

u - \frac{1}{u} - (u + \frac{1}{u}) = 0

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}

u - \frac{1}{u} - u - \frac{1}{u} = 0

Multiply both sides by

u

u - \frac{1}{u} - u - \frac{1}{u} = 0

Multiply both sides by

u

uu - \frac{1}{u} u - uu - \frac{1}{u} u = 0 \cdot u

Simplify

uu - \frac{1}{u} u - uu - \frac{1}{u} u = 0 \cdot 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

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

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} - 1 - u^{2} - 1 = 0

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

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

Group like terms

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

Add similar elements:

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

= - 1 - 1

Subtract the numbers:

- 1 - 1 = - 2

= - 2

- 2 = 0

- 2 = 0

- 2 = 0

The sides are not equal

No Solution for u \in R

No Solution for z \in R

No Solution for z \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(z)-cosh(z)=0 ?
The general solution for sinh(z)-cosh(z)=0 is No Solution for z\in\mathbb{R}