What is the general solution for tanh(x)=0 ?

The general solution for tanh(x)=0 is x=0

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tanh(x)=0

Solution

tanh (x) = 0

Solution

x = 0

Degrees

x = 0^{\circ}

Solution steps

tanh (x) = 0

Rewrite using trig identities

tanh (x) = 0

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0 : x = 0

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

e^{x} - e^{- x} = 0

Add

e^{- x}

to both sides

e^{x} - e^{- x} + e^{- x} = 0 + e^{- x}

Simplify

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

Apply exponent rules

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

a^{f (x)} = a^{g (x)}

, then

f (x) = g (x)

x = - x

x = - x

Solve

x = - x : x = 0

x = - x

Move

x

to the left side

x = - x

Add

x

to both sides

x + x = - x + x

Simplify

2 x = 0

2 x = 0

Divide both sides by

2

2 x = 0

Divide both sides by

2

\frac{2 x}{2} = \frac{0}{2}

Simplify

x = 0

x = 0

x = 0

Verify Solutions:

x = 0

True

Check the solutions by plugging them into

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0

Remove the ones that don't agree with the equation.

Plug in

x = 0 :

True

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

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

\frac{e ^{0} - e ^{- 0}}{e ^{0} + e ^{- 0}}

Apply rule

a^{0} = 1, a \neq = 0

e^{0} = 1, e^{- 0} = 1

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

Subtract the numbers:

1 - 1 = 0

= \frac{0}{1 + 1}

Add the numbers:

1 + 1 = 2

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

0 = 0

True

The solution is

x = 0

x = 0

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Frequently Asked Questions (FAQ)

What is the general solution for tanh(x)=0 ?
The general solution for tanh(x)=0 is x=0