Answer
The six trigonometric functions of $\theta $ for point $\left( 3,7 \right)$ are,
$\sin \theta =\frac{7\sqrt{58}}{58},\cos \theta =\frac{3\sqrt{58}}{58},\tan \theta =\frac{7}{3},\csc \theta =\frac{\sqrt{58}}{7},\sec \theta =\frac{\sqrt{58}}{3}$ and $\cot \theta =\frac{3}{7}$.
Work Step by Step
Consider the point $\left( 3,7 \right)$. Here, $x=3$ and $y=7$
The six trigonometric functions of $\theta $ are defined in the term of ratios.
According to the Pythagoras theorem, the hypotenuse is,
$r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
Substitute $3$ for $x$ and $7$ for $y$.
$\begin{align}
& r=\sqrt{{{\left( 3 \right)}^{2}}+{{\left( 7 \right)}^{2}}} \\
& =\sqrt{9+49} \\
& =\sqrt{58}
\end{align}$
Write the trigonometric expression of $\sin \theta $.
$\sin \theta =\frac{y}{r}$
Substitute $7$ for $y$ and $\sqrt{58}$ for $r$.
$\begin{align}
& \sin \theta =\frac{7}{\sqrt{58}} \\
& =\frac{7}{\sqrt{58}}.\frac{\sqrt{58}}{\sqrt{58}} \\
& =\frac{7\sqrt{58}}{58}
\end{align}$
Recall the trigonometric expression of $\cos \theta $.
$\cos \theta =\frac{x}{r}$
Substitute $3$ for $x$ and $\sqrt{58}$ for $r$.
$\begin{align}
& \cos \theta =\frac{3}{\sqrt{58}} \\
& =\frac{3}{\sqrt{58}}.\frac{\sqrt{58}}{\sqrt{58}} \\
& =\frac{3\sqrt{58}}{58}
\end{align}$
Recall the trigonometric expression of $\tan \theta $.
$\tan \theta =\frac{y}{x}$
Substitute $3$ for $x$ and $7$ for $y$.
$\tan \theta =\frac{7}{3}$
Recall the trigonometric expression of $\csc \theta $.
$\csc \theta =\frac{r}{y}$
Substitute $7$ for $y$ and $\sqrt{58}$ for $r$.
$\csc \theta =\frac{\sqrt{58}}{7}$
Recall the trigonometric expression of $\sec \theta $.
$\sec \theta =\frac{r}{x}$
Substitute $3$ for $x$ and $\sqrt{58}$ for $r$.
$\sec \theta =\frac{\sqrt{58}}{3}$
Recall the trigonometric expression of $\cot \theta $.
$\cot \theta =\frac{x}{y}$
Substitute $3$ for $x$ and $7$ for $y$.
$\cot \theta =\frac{3}{7}$
Thus, the six trigonometric functions of $\theta $ for point $\left( 3,7 \right)$ are,
$\sin \theta =\frac{7\sqrt{58}}{58},\cos \theta =\frac{3\sqrt{58}}{58},\tan \theta =\frac{7}{3},\csc \theta =\frac{\sqrt{58}}{7},\sec \theta =\frac{\sqrt{58}}{3}$ and $\cot \theta =\frac{3}{7}$.
