Which of the following is correct?

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(A) Using the exponential definition of trigonometric functions,we have $\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$.Substituting $z = ix$,we get $\sin(ix)

Vedclass · Accepted Answer

$\sin(ix) = i\sinh x$

Vedclass · Answer

(A) Using the exponential definition of trigonometric functions,we have $\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$.Substituting $z = ix$,we get $\sin(ix) = \frac{e^{i(ix)} - e^{-i(ix)}}{2i} = \frac{e^{-x} - e^x}{2i}$.Since $\sinh x = \frac{e^x - e^{-x}}{2}$,we can write $\sin(ix) = \frac{-(e^x - e^{-x})}{2i} = -\frac{1}{i} \sinh x$.Multiplying the numerator and denominator by $i$,we get $\sin(ix) = -\frac{i}{i^2} \sinh x = -\frac{i}{-1} \sinh x = i\sinh x$.Thus,the correct identity is $\sin(ix) = i\sinh x$.

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