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

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

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Popular Trigonometry >

tanh(z)=0

Solution

tanh (z) = 0

Solution

z = 0

Degrees

z = 0^{\circ}

Solution steps

tanh (z) = 0

Rewrite using trig identities

tanh (z) = 0

Use the Hyperbolic identity:

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

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

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

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

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

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

e^{z} - e^{- z} = 0

Add

e^{- z}

to both sides

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

Simplify

e^{z} = e^{- z}

Apply exponent rules

e^{z} = e^{- z}

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

, then

f (x) = g (x)

z = - z

z = - z

Solve

z = - z : z = 0

z = - z

Move

z

to the left side

z = - z

Add

z

to both sides

z + z = - z + z

Simplify

2 z = 0

2 z = 0

Divide both sides by

2

2 z = 0

Divide both sides by

2

\frac{2 z}{2} = \frac{0}{2}

Simplify

z = 0

z = 0

z = 0

Verify Solutions:

z = 0

True

Check the solutions by plugging them into

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

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

Plug in

z = 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

z = 0

z = 0

Graph

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Frequently Asked Questions (FAQ)

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