What is the general solution for sinh(npi)=0 ?

The general solution for sinh(npi)=0 is No Solution for n\in\mathbb{R}

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sinh(npi)=0

sinh (nπ) = 0

No Solution for n \in R

Solution steps

sinh (nπ) = 0

Rewrite using trig identities

sinh (nπ) = 0

Use the Hyperbolic identity:

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

\frac{e ^{nπ} - e ^{- nπ}}{2} = 0

\frac{e ^{nπ} - e ^{- nπ}}{2} = 0

\frac{e ^{nπ} - e ^{- nπ}}{2} = 0 : n = 0

\frac{e ^{nπ} - e ^{- nπ}}{2} = 0

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

e^{nπ} - e^{- nπ} = 0

Add

e^{- nπ}

to both sides

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

Simplify

e^{nπ} = e^{- nπ}

Apply exponent rules

e^{nπ} = e^{- nπ}

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

, then

f (x) = g (x)

nπ = - nπ

nπ = - nπ

Solve

nπ = - nπ : n = 0

nπ = - nπ

Move

nπ

to the left side

nπ = - nπ

Add

nπ

to both sides

nπ + nπ = - nπ + nπ

Simplify

2 πn = 0

2 πn = 0

Divide both sides by

2 π

2 πn = 0

Divide both sides by

2 π

\frac{2 πn}{2 π} = \frac{0}{2 π}

Simplify

n = 0

n = 0

n = 0

n = 0

Since the equation is undefined for:

0

No Solution for n \in R

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