But z is already given in rectangular form... A complex number in rectangular form looks like [tex]a+bi[/tex], where [tex]a,b\in\mathbb{R}[/tex].
Perhaps you're supposed to write cos(12º) and sin(12º) in non-trigonometric form? In that case, we want exact forms of these numbers.
Note that 12º = 60º/5. Consider the identities
[tex]\cos(5t)=\cos^5t-10\sin^2t\cos^3t+5\sin^4t\cos t[/tex]
[tex]\implies\cos(5t)=16\cos^5t-20\cos^3t+5\cos t[/tex]
[tex]\sin(5t)=\sin^5t-10\sin^3t\cos^2t+5\sin t\cos^4t[/tex]
[tex]\implies\sin(5t)=16\sin^5t-20\sin^3t+5\sin t[/tex]
(both of which follow from DeMoivre's theorem)
We have [tex]\sin(60^\circ)=\frac{\sqrt3}2[/tex] and [tex]\cos(60^\circ)=\frac12[/tex], so we get
[tex]\cos(12^\circ)=\dfrac{\sqrt5-1+\sqrt{30+6\sqrt5}}8[/tex]
[tex]\sin(12^\circ)=\dfrac{\sqrt{7-\sqrt5-\sqrt{30-6\sqrt5}}}4[/tex]
