Answer
a) The value of $\cos \theta $ is $\frac{x}{r}$.
b) The value of the ratio is $\frac{-3\sqrt{34}}{34}$ and it is negative.
Work Step by Step
(a)
We have to write the expression for $\cos \theta $ as follows:
$\cos \theta =\frac{\text{side adjacent to }\theta }{\text{hypotenuse}}$
In figure (a), $\text{the side adjacent to }\theta $ is $x$ and the $\text{hypotenuse}$ is $r$.
Hence, the value of $\cos \theta $ is
$\cos \theta =\frac{x}{r}$
(b)
So, according to the Pythagoras theorem
$\begin{align}
& {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\
& r=\sqrt{{{x}^{2}}+{{y}^{2}}} \\
\end{align}$
Put $-3$ for $x$ and $5$ for $y$ as follows:
$\begin{align}
& r=\sqrt{{{\left( -3 \right)}^{2}}+{{\left( 5 \right)}^{2}}} \\
& =\sqrt{9+25} \\
& =\sqrt{34}
\end{align}$
The ratio determined in part (a) is
$\frac{x}{r}$
Put $-3$ for $x$ and $\sqrt{34}$ for $r$ as follows:
$\frac{x}{r}=\frac{-3}{\sqrt{34}}$
And multiply and divide the denominator as follows:
$\begin{align}
& \frac{x}{r}=\frac{-3}{\sqrt{34}}\times \frac{\sqrt{34}}{\sqrt{34}} \\
& =\frac{-3\sqrt{34}}{\left( {{\sqrt{34}}^{2}} \right)} \\
& =\frac{-3\sqrt{34}}{34}
\end{align}$
