I’m self studying through Stroud’s Engineering Mathematics 7th Ed. and can’t seem to figure out how one of the answers was arrived at.
Here is the question:
“Obtain the expansion of $\sin(x-iy)$ in terms of the trigonometric and hyperbolic functions of $x$ and $y$.”
Using trig identities I get:
$$\sin(x)\cos(iy) - \cos(x)\sin(iy)$$
Then I use $\cos(iy) = \cosh(y)$, $\sin(iy) = i\sinh(y)$ and $\cos(x) = \cosh(ix)$
Which gives me:
$$\sin(x)\cosh(y) - \cosh(ix)i\sinh(y)$$
However, the solution provided is $\sin(x)\cosh(y) - i\cos(x)\sinh(y)$
I’ve looked up errata in case it’s incorrect, but it’s not mentioned in the errata, so I suspect I’ve made an error somewhere.
Hoping someone can shed some light.
Thanks.
By definition, $$\sin(z)=\frac{i}{2}(e^{-iz}-e^{iz})$$ Thus $$\sin(x-iy)=\frac{i}{2}(e^{-y-ix}-e^{y+ix})$$ $$=\frac{i}{2}(e^{-y}(\cos(x)-i\sin(x))-e^y(\cos(x)+i\sin(x)))$$ You can take it from there.