Unit step function value

Question

I have understand question c)(i) But i do not understand question c)(ii)

Answer 1

The unit step function is defined as

$$u(t-a) = \left\{\begin{matrix} 1 & t \geq a \\ 0& 0 \leq t < a \end{matrix}\right.$$

So for instance in your case $a = 1$ and plugging $t = \frac{5\pi}{6}$ in gives

$$u\left(\frac{5\pi}{6} - 1 \right) = \left\{\begin{matrix} 1 & t \geq 1 \\ 0& 0\leq t < 1 \end{matrix}\right.$$

Note that $t = 5\pi/6 > 1$, therefore the value of $u\left(\frac{5\pi}{6} - 1 \right) = 1.$

Since the defintion of $g(t)$ has been worked out, we find that $$g \left ( \frac{5\pi}{6}\right) u\left(\frac{5\pi}{6} - 1 \right) = \left( \sin \frac{5\pi}{6} \right ) (1) = 1/2$$ because our value of $t$ lies between $$\frac{\pi}{2} \leq \frac{5\pi}{6} \leq \frac{3\pi}{2}.$$