Let $z = \sin(4t)+i(1-\cos(4t)), t \in [0,\pi/4)$; find $|z|$ and $\text {arg} (z)$

Question

Let $z = \sin(4t)+i(1-\cos(4t)), t \in [0,\pi/4)$ Find the modulus $|z|$ and the principal argument $\text {arg} (z)$ . Express your solution in terms of $t$.

Answer 1

Hint:

$$z=2\sin(2t)\cos(2t)+2i\sin^2(2t)=2\sin(2t)(\cos(2t)+i\sin(2t))=2\sin(2t)e^{i2t}$$ Using:

$\sin(2\theta)=2\sin(\theta)\cos(\theta)$ and $1-\cos(2\theta)=2\sin^2(\theta)$

Answer 2

Hint: Let $a = \sin 4t$ and $b = 1 - \cos 4t$.

The modulus of $z$ is $|z|$ = $\sqrt {a^2 + b^2}$; the argument of $z$ is $\text {arg } z = \arctan (b/a)$.

You should get $|z| = 2 \sin 2t$ and $\text {arg } z = 2t$ once you generate the work.