The progress I have made is as follows:
$\sin(t)$ is our signal therefore $\omega = 1 = 2\pi f$ and $f$ = $\frac{1}{2 \pi}$
Also, $f_s$ = 10Hz therefore T = $\frac{1}{f_s}$ = 0.1s
$H(z) = \frac{z}{z-0.7}$ so let $z = e^{j\omega T}$ where $\omega T = 1 \times0.1 = 0.1$
$H(\omega) = \frac{e^{j0.1}}{e^{j0.1} - 0.7}$
Taking the denominator:
$\cos(0.1) + j\sin(0.1) = 0.995 + 0.099j$
therefore,
$\cos(0.1) -0.7 + j\sin(0.1) = 0.311∠0.324$
$H(\omega) = \frac{1∠0.1}{0.311∠0.324}$
$H(\omega) = 3.21∠0.224$
From here I am out of ideas on how to continue. Any advice appreciated.