Write cos(2+i) in the form u+iv

Write $\cos(2+i)$ in the form $u+iv$.

Step 1 · Apply with $x=2,\ y=1$

$$ \cos(2+i)=\cos(2)\cosh(1)-i\sin(2)\sinh(1) $$

Step 2 · Numerical values

$$ \cos(2)\approx-0.416147,\quad \sin(2)\approx0.909297,\quad \cosh(1)=\frac{e+e^{-1}}2\approx1.543081,\quad \sinh(1)=\frac{e-e^{-1}}2\approx1.175201 $$

Step 3 · Real and imaginary parts

$$ u=\cos(2)\cosh(1)\approx-0.642148, \qquad v=-\sin(2)\sinh(1)\approx-1.068607 $$

Step 4 · Result

$$ \boxed{\cos(2+i)=\cos(2)\cosh(1)-i\sin(2)\sinh(1)\approx-0.6421-1.0686i} $$

Verification

Via $\cos z=\frac{e^{iz}+e^{-iz}}2$ with $z=2+i$: $\ iz=-1+2i$, so $e^{iz}=e^{-1}(\cos2+i\sin2)$ and $e^{-iz}=e(\cos2-i\sin2)$; averaging gives $\cos2\cdot\frac{e+e^{-1}}2-i\sin2\cdot\frac{e-e^{-1}}2=\cos2\cosh1-i\sin2\sinh1$, matching Step 1 exactly.