We know that $$\sin(a\pm b)=\cos(b)\sin(a)\pm \cos(a)\sin(b)$$
We also know that $$\sinh(x)=-i\,\sin(ix) \quad\text{and}\quad\cosh(x)=\cos(ix)$$
Now, suppose I want to compute $\sin(5-8i)$.
Then, $$\sin(5-8i)=\cos(8i)\sin(5)-\sin(8i)\cos(5) \tag{1}$$
Now we substitute trigonometric definitions of hyperbolic functions in (1). Then, we get,
$$\begin{align} \sin(5-8i) &=\cosh(8)\sin(5)-\cos(5)(-i\,\sinh(8)) \\ &=\cosh(8)\sin(5)+i\,\cos(5)\sinh(8) \end{align}\tag{2}$$
My hp 50g calculator gives answer to $\sin{(5-8i)}$ in radian mode, $$(-1429.2566486,-422.79248111)$$ But R.H.S. of $(2)$ is $$(-1429.2566486,422.79248111)$$
Where am I wrong in this computation?
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