trigonometry - Real and imaginary part from trigonometric form

In school I am learning for complex numbers in trigonometric form.
$$z= a + bi r ( \cos(\alpha) + i\sin(\alpha) )$$
In a problem I have to find the real and imaginary part from trigonometric form of
$$1 + \cos(\alpha) + i\sin(\alpha)$$
For which I think the solution is

• Real part = $1+ \arccos(\alpha)$


• Imaginary part = $\arcsin(\alpha)$

PR2: $\sin(\alpha)+i\cos(\alpha)$
Update>
For which I think the solution is>

• Real part = $\arcsin(\alpha)$


• Imaginary part = $\arccos(\alpha)$