What is the value of cosh(pii) ?

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cosh(pii)

cosh (πi)

- 1

Solution steps

cosh (πi)

Use the Hyperbolic identity:

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

= \frac{e ^{iπ} + e ^{- iπ}}{2}

\frac{e ^{iπ} + e ^{- iπ}}{2} = - 1

\frac{e ^{iπ} + e ^{- iπ}}{2}

e^{iπ} + e^{- iπ} = - 2

e^{iπ} + e^{- iπ}

Apply imaginary number rule:

e^{iπ} = - 1

= - 1 + e^{- iπ}

e^{- iπ} = - 1

e^{- iπ}

Apply imaginary number rule:

e^{iaπ} = (- 1)^{a}

= (- 1)^{- 1}

Apply exponent rule:

(- a)^{n} = - a^{n},

n

is odd

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

= - 1^{- 1}

Apply rule

1^{a} = 1

= - 1

= - 1 - 1

Subtract the numbers:

- 1 - 1 = - 2

= - 2

= \frac{- 2}{2}

Simplify

\frac{- 2}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

= - 1

= - 1

9 sin (0)

arccos (tan (\frac{5 π}{4}))

tan (16 0^{\circ})

\frac{4}{sin ( 3 0 ^{\circ} )}

\frac{12}{sin ( 2 9 ^{\circ} )}