Answer
a) The ratio that expresses $\sin \theta $ is $\frac{y}{r}$.
b) The ratio for figure (b) is $\frac{y}{r}=\frac{4}{5}$, that is, positive.
Work Step by Step
(a)
In this figure, both x and y coordinates are positive as they lie in the first quadrant; the ordinate is y and abscissa is x.
$\sin \theta =\frac{y}{r}$
(b)
In the second figure, we calculate the value of r as follows:
$\begin{align}
& r=\sqrt{{{x}^{2}}+{{y}^{2}}} \\
& r=\sqrt{{{\left( -3 \right)}^{2}}+{{4}^{2}}} \\
& r=\sqrt{9+16} \\
& r=\sqrt{25} \\
& r=5 \\
& \frac{y}{r}=\frac{4}{5}
\end{align}$
Thus, the above ratio obtained is positive.
