Intro to hyperbolic functions

Algebra Level pending

Evaluate cos ⁡ ( a i ) \cos{(ai)} where a a is a real number and i = − 1 i = \sqrt{-1} .

For more information, read hyperbolic trigonometric functions .

coth ⁡ ( a ) \coth{(a)} sinh ⁡ ( a ) \sinh{(a)} sinh ⁡ ( 2 a ) \sinh{(2a)} None of the answer choices are correct tanh ⁡ ( a ) \tanh{(a)} cosh ⁡ ( a ) \cosh{(a)}

Akeel Howell by Akeel Howell , 22, USA

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1 solution

Akeel Howell
Mar 16, 2017

Relevant wiki: Hyperbolic Trigonometric Functions

cos ⁡ x \cos{x} can be rewritten as e i x + e − i x 2 \large\dfrac{e^{ix}+e^{-ix}}{2} ⟹ cos ⁡ ( a i ) = \implies \cos{(ai)} =\ e i ( a i ) + e − i ( a i ) 2 \large\dfrac{e^{i(ai)}+e^{-i(ai)}}{2}

⟹ e − a + e a 2 = cosh ⁡ a \implies \dfrac{e^{-a}+e^{a}}{2} = \cosh{a} .

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