## Precalculus (6th Edition) Blitzer

The values of the six trigonometric functions of $\theta$ are $\sin \theta =\frac{5\sqrt{89}}{89},\cos \theta =\frac{8\sqrt{89}}{89},\tan \theta =\frac{5}{8},\csc \theta =\frac{\sqrt{89}}{5},\sec \theta =\frac{\sqrt{89}}{8}\text{,}\cot \theta =\frac{8}{5}$
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}