use the information contained in the figure to determine the values of the six trigonometric functions of θ…

Explanation:

Step1: Find the other side using Pythagorean theorem

Let the unknown side be $y$. By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 8\sqrt{2}$ and $a = 8$. Then $y=\sqrt{(8\sqrt{2})^{2}-8^{2}}=\sqrt{128 - 64}=\sqrt{64}=8$.

Step2: Define sine function

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{8}{8\sqrt{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

Step3: Define cosine function

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{8}{8\sqrt{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

Step4: Define tangent function

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{8}{8}=1$

Step5: Define cosecant function

$\csc\theta=\frac{1}{\sin\theta}=\sqrt{2}$

Step6: Define secant function

$\sec\theta=\frac{1}{\cos\theta}=\sqrt{2}$

Step7: Define cotangent function

$\cot\theta=\frac{1}{\tan\theta}=1$

Answer:

$\sin\theta=\frac{\sqrt{2}}{2}$, $\cos\theta=\frac{\sqrt{2}}{2}$, $\tan\theta = 1$, $\csc\theta=\sqrt{2}$, $\sec\theta=\sqrt{2}$, $\cot\theta = 1$