Answer
The six trigonometric functions of $\theta $ for point $\left( 5,-5 \right)$ are,
$\sin \theta =-\frac{\sqrt{2}}{2},\cos \theta =\frac{\sqrt{2}}{2},\tan \theta =-1,\csc \theta =-\sqrt{2},\sec \theta =\sqrt{2}$ and $\cot \theta =-1$.
Work Step by Step
Consider the point $\left( 5,-5 \right)$. Here, $x=5$ and $y=-5$.
The six trigonometric functions of $\theta $ are defined in terms of a ratio.
According to the Pythagoras theorem, the hypotenuse is,
$r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
Substitute $5$ for $x$ and $-5$ for $y$.
$\begin{align}
& r=\sqrt{{{\left( 5 \right)}^{2}}+{{\left( -5 \right)}^{2}}} \\
& =\sqrt{25+25} \\
& =\sqrt{50} \\
& =5\sqrt{2}
\end{align}$
Recall the trigonometric expression of $\sin \theta $.
$\sin \theta =\frac{y}{r}$
Substitute $-5$ for $y$ and $5\sqrt{2}$ for $r$.
$\begin{align}
& \sin \theta =-\frac{5}{5\sqrt{2}} \\
& =-\frac{1}{\sqrt{2}} \\
& =-\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}} \\
& =\frac{\sqrt{2}}{2}
\end{align}$
Recall the trigonometric expression of $\cos \theta $.
$\cos \theta =\frac{x}{r}$
Substitute $5$ for $x$ and $5\sqrt{2}$ for $r$.
$\begin{align}
& \cos \theta =\frac{5}{5\sqrt{2}} \\
& =\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}} \\
& =\frac{\sqrt{2}}{2}
\end{align}$
Recall the trigonometric expression of $\tan \theta $.
$\tan \theta =\frac{y}{x}$
Substitute $5$ for $x$ and $-5$ for $y$.
$\begin{align}
& \tan \theta =\frac{-5}{5} \\
& =-1
\end{align}$
Recall the trigonometric expression of $\csc \theta $.
$\csc \theta =\frac{r}{y}$
Substitute $-5$ for $y$ and $5\sqrt{2}$ for $r$.
$\begin{align}
& \csc \theta =\frac{5\sqrt{2}}{-5} \\
& =-\sqrt{2}
\end{align}$
Recall the trigonometric expression of $\sec \theta $.
$\sec \theta =\frac{r}{x}$
Substitute $5$ for $x$ and $5\sqrt{2}$ for $r$.
$\begin{align}
& \sec \theta =\frac{5\sqrt{2}}{5} \\
& =\sqrt{2}
\end{align}$
Recall the trigonometric expression of $\cot \theta $.
$\cot \theta =\frac{x}{y}$
Substitute $5$ for $x$ and $-5$ for $y$.
$\begin{align}
& \cot \theta =\frac{5}{-5} \\
& =-1
\end{align}$
The six trigonometric functions of $\theta $ for point $\left( 5,-5 \right)$ are,
$\sin \theta =-\frac{\sqrt{2}}{2},\cos \theta =\frac{\sqrt{2}}{2},\tan \theta =-1,\csc \theta =-\sqrt{2},\sec \theta =\sqrt{2}$ and $\cot \theta =-1$.
