Q.Express the following number in the form $x+iy$ : $\sqrt2\left(\cos\dfrac{7\pi}{4}+i\sin\dfrac{7\pi}{4}\right)$

Writing the coordinates $a=r\cos\theta$ and $b=r\sin\theta$ of the point representing $z=a+ib$ in terms of its modulus $r=|z|$ and argument $\theta=\arg z$ turns the standard form into the polar form $z=r(\cos\theta+i\sin\theta)$, with $P(r,\theta)$ called the polar coordinates of the point and the origin renamed the pole. Using Euler's identity $e^{i\theta}=\cos\theta+i\sin\theta$ (a deeper result proved using infinite series, stated here without proof), the same complex number can be written even more compactly in exponential form as $z=re^{i\theta}$. Both forms package the same two pieces of information — how far the point is from the origin ($r$) and in which direction ($\theta$) — and they are especially convenient for multiplication, division and raising to powers, since multiplying two exponential-form numbers just multiplies their $r$'s and adds their exponents' angles: $\left(r_1e^{i\theta_1}\right)\left(r_2e^{i\theta_2}\right)=r_1r_2\,e^{i(\theta_1+\theta_2)}$, …