Answer
The length of the side is $b=7$ and the values of the six trigonometric functions are $\sin \theta =\frac{24}{25}$ , $\cos \theta =\frac{7}{25}$ , $\tan \theta =\frac{24}{7}$ , $\csc \theta =\frac{25}{24}$ , $\sec \theta =\frac{25}{7}$ and $\cot \theta =\frac{7}{24}$.
Work Step by Step
In the right angle triangle, $a=24$ , $b$ is the side adjacent with angle $\theta $ and $c=25$.
According to the Pythagoras theorem,
${{a}^{2}}+{{b}^{2}}={{c}^{2}}$
Rearrange for $b$.
$\begin{align}
& {{b}^{2}}={{c}^{2}}-{{a}^{2}} \\
& b=\sqrt{{{c}^{2}}-{{a}^{2}}} \\
\end{align}$
Substitute $25$ for $c$ and $24$ for $a$.
$\begin{align}
& b=\sqrt{{{\left( 25 \right)}^{2}}-{{\left( 24 \right)}^{2}}} \\
& =\sqrt{625-576} \\
& =\sqrt{49} \\
& =7
\end{align}$
The ratio of $\sin \theta $ is
$\sin \theta =\frac{a}{c}$
Substitute $25$ for $c$ and $24$ for $a$.
$\sin \theta =\frac{24}{25}$
The ratio of $\cos \theta $ is
$\cos \theta =\frac{b}{c}$
Substitute $25$ for $c$ and $7$ for $b$.
$\cos \theta =\frac{7}{25}$
The ratio of $\tan \theta $ is
$\tan \theta =\frac{a}{b}$
Substitute $24$ for $a$ and $7$ for $b$.
$\tan \theta =\frac{24}{7}$
The ratio of $\csc \theta $ is
$\csc \theta =\frac{c}{a}$
Substitute $25$ for $c$ and $24$ for $a$.
$\csc \theta =\frac{25}{24}$
The ratio of $\sec \theta $ is
$\sec \theta =\frac{c}{b}$
Substitute $25$ for $c$ and $7$ for $b$.
$\sec \theta =\frac{25}{7}$
The ratio of $\cot \theta $ is
$\cot \theta =\frac{b}{a}$
Substitute $24$ for $a$ and $7$ for $b$.
$\cot \theta =\frac{7}{24}$
Therefore, the length of the side is $b=7$ and the values of the six trigonometric functions are, $\sin \theta =\frac{24}{25}$ , $\cos \theta =\frac{7}{25}$ , $\tan \theta =\frac{24}{7}$ , $\csc \theta =\frac{25}{24}$ , $\sec \theta =\frac{25}{7}$ and $\cot \theta =\frac{7}{24}$.
