What is the general solution for tanh(x)=1 ?

The general solution for tanh(x)=1 is No Solution for x\in\mathbb{R}

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Popular >

tanh(x)=1

Solution

tanh (x) = 1

Solution

No Solution for x \in R

Solution steps

tanh (x) = 1

Rewrite using trig identities

tanh (x) = 1

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 1

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 1

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 1 :

No Solution for

x \in R

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 1

Multiply both sides by

e^{x} + e^{- x}

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} (e^{x} + e^{- x}) = 1 \cdot (e^{x} + e^{- x})

Simplify

e^{x} - e^{- x} = e^{x} + e^{- x}

Apply exponent rules

e^{x} - e^{- x} = e^{x} + e^{- x}

Apply exponent rule:

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

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

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

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

Rewrite the equation with

e^{x} = u

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

Solve

u - u^{- 1} = u + u^{- 1} :

No Solution for

u \in R

u - u^{- 1} = u + u^{- 1}

Refine

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

Subtract

u

from both sides

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

Simplify

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

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

\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

- 1 = 1

- 1 = 1

- 1 = 1

The sides are not equal

No Solution for u \in R

No Solution for x \in R

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for tanh(x)=1 ?
The general solution for tanh(x)=1 is No Solution for x\in\mathbb{R}