Answer
The six trigonometric functions are:
$\sin \theta =-\frac{5\sqrt{26}}{26}\text{; cos}\theta \text{=}-\frac{\sqrt{26}}{26}\text{; tan}\theta \text{=5; csc}\theta \text{=}-\frac{\sqrt{26}}{5};\text{ sec}\theta \text{=}-\sqrt{26};\text{ cot}\theta \text{=}\frac{1}{5}$.
Work Step by Step
Here, $ x=-1\text{ and }y=-5$. Then, $\begin{align}
& r=\sqrt{{{\left( -1 \right)}^{2}}+{{\left( -5 \right)}^{2}}} \\
& =\sqrt{1+25} \\
& =\sqrt{26}
\end{align}$
Now, evaluate the sine functions as follows:
$\begin{align}
& \sin \theta =\frac{y}{r} \\
& =\frac{-5}{\sqrt{26}} \\
& =\frac{-5\sqrt{26}}{\sqrt{26}\cdot \sqrt{26}} \\
& =-\frac{5\sqrt{26}}{26}
\end{align}$
Then, evaluate the cosine functions as follows:
$\begin{align}
& \cos \theta =\frac{x}{r} \\
& =\frac{-1}{\sqrt{26}} \\
& =\frac{-1\sqrt{26}}{\sqrt{26}\cdot \sqrt{26}} \\
& =-\frac{\sqrt{26}}{26}
\end{align}$
Then, evaluate the tangent functions as follows:
$\begin{align}
& \tan \theta =\frac{y}{x} \\
& =\frac{-5}{-1} \\
& =5
\end{align}$
Then, evaluate the cosecant functions as follows:
$\begin{align}
& \csc \theta =\frac{r}{y} \\
& =\frac{\sqrt{26}}{-5} \\
& =-\frac{\sqrt{26}}{5}
\end{align}$
Then evaluate the secant functions as follows:
$\begin{align}
& \sec \theta =\frac{r}{x} \\
& =\frac{\sqrt{26}}{-1} \\
& =-\sqrt{26}
\end{align}$
And finally evaluate the cotangent functions as follows:
$\begin{align}
& \cot \theta =\frac{x}{y} \\
& =\frac{-1}{-5} \\
& =\frac{1}{5}
\end{align}$
