Answer
The values of the six trigonometric functions of $\theta $ are
$\sin \theta =\frac{5\sqrt{89}}{89},\cos \theta =\frac{8\sqrt{89}}{89},\tan \theta =\frac{5}{8},\csc \theta =\frac{\sqrt{89}}{5},\sec \theta =\frac{\sqrt{89}}{8}\text{,}\cot \theta =\frac{8}{5}$
Work Step by Step
By using the Pythagorean Theorem, calculate the hypotenuse AB as show below:
$\begin{align}
& AB=\sqrt{A{{C}^{2}}+B{{C}^{2}}} \\
& =\sqrt{{{8}^{2}}+{{5}^{2}}} \\
& =\sqrt{89}
\end{align}$
Now, calculate the sine function as follows:
$\begin{align}
& \sin \theta =\frac{BC}{AB} \\
& =\frac{5}{\sqrt{89}} \\
& =\frac{5}{\sqrt{89}}\times \frac{\sqrt{89}}{\sqrt{89}} \\
& =\frac{5\sqrt{89}}{89}
\end{align}$
Then, calculate the cosine function as follows:
$\begin{align}
& \cos \theta =\frac{AC}{AB} \\
& =\frac{8}{\sqrt{89}} \\
& =\frac{8}{\sqrt{89}}\times \frac{\sqrt{89}}{\sqrt{89}} \\
& =\frac{8\sqrt{89}}{89}
\end{align}$
Then, calculate the tangent function as follows:
$\begin{align}
& \tan \theta =\frac{BC}{AC} \\
& =\frac{5}{8}
\end{align}$
Then, calculate the cosecant function as follows:
$\begin{align}
& \csc \theta =\frac{AB}{BC} \\
& =\frac{\sqrt{89}}{5}
\end{align}$
Then, calculate the secant function as follows:
$\begin{align}
& \sec \theta =\frac{AB}{AC} \\
& =\frac{\sqrt{89}}{8}
\end{align}$
And finally calculate the cotangent function as follows:
$\begin{align}
& \cot \theta =\frac{AC}{BC} \\
& =\frac{8}{5}
\end{align}$
