Answer
The six trigonometric functions of $\theta $ for point $\left( 2,3 \right)$ are,
$\sin \theta =\frac{3\sqrt{13}}{13},\cos \theta =\frac{2\sqrt{13}}{13},\tan \theta =\frac{3}{2},\csc \theta =\frac{\sqrt{13}}{3},\sec \theta =\frac{\sqrt{13}}{2}$ and $\cot \theta =\frac{2}{3}$.
Work Step by Step
According to the Pythagoras theorem in a right angle triangle, the hypotenuse is given as,
${{r}^{2}}={{x}^{2}}+{{y}^{2}}$
Consider the point $\left( 2,3 \right)$. Here, $x=2$ and $y=3$.
The six trigonometric functions of $\theta $ are defined in terms of the ratio.
According to the Pythagoras theorem, the hypotenuse is,
$r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
Substitute $2$ for $x$ and $3$ for $y$.
$\begin{align}
& r=\sqrt{{{\left( 2 \right)}^{2}}+{{\left( 3 \right)}^{2}}} \\
& =\sqrt{4+9} \\
& =\sqrt{13}
\end{align}$
Recall the trigonometric expression of $\sin \theta $.
$\sin \theta =\frac{y}{r}$
Substitute $3$ for $y$ and $\sqrt{13}$ for $r$.
$\begin{align}
& \sin \theta =\frac{3}{\sqrt{13}} \\
& =\frac{3}{\sqrt{13}}.\frac{\sqrt{13}}{\sqrt{13}} \\
& =\frac{3\sqrt{13}}{13}
\end{align}$
Recall the trigonometric expression of $\cos \theta $.
$\cos \theta =\frac{x}{r}$
Substitute $2$ for $x$ and $\sqrt{13}$ for $r$.
$\begin{align}
& \cos \theta =\frac{2}{\sqrt{13}} \\
& =\frac{2}{\sqrt{13}}.\frac{\sqrt{13}}{\sqrt{13}} \\
& =\frac{2\sqrt{13}}{13}
\end{align}$
Recall the trigonometric expression of $\tan \theta $.
$\tan \theta =\frac{y}{x}$
Substitute $2$ for $x$ and $3$ for $y$.
$\tan \theta =\frac{3}{2}$
Recall the trigonometric expression of $\csc \theta $.
$\csc \theta =\frac{r}{y}$
Substitute $3$ for $y$ and $\sqrt{13}$ for $r$.
$\csc \theta =\frac{\sqrt{13}}{3}$
Recall the trigonometric expression of $\sec \theta $.
$\sec \theta =\frac{r}{x}$
Substitute $2$ for $x$ and $\sqrt{13}$ for $r$.
$\sec \theta =\frac{\sqrt{13}}{2}$
Recall the trigonometric expression of $\cot \theta $.
$\cot \theta =\frac{x}{y}$
Substitute $2$ for $x$ and $3$ for $y$.
$\cot \theta =\frac{2}{3}$
Thus, the six trigonometric functions of $\theta $ for point $\left( 2,3 \right)$ are,
$\sin \theta =\frac{3\sqrt{13}}{13},\cos \theta =\frac{2\sqrt{13}}{13},\tan \theta =\frac{3}{2},\csc \theta =\frac{\sqrt{13}}{3},\sec \theta =\frac{\sqrt{13}}{2}$ and $\cot \theta =\frac{2}{3}$.
