find the exact values of the six trigonometric functions of the angle $\\theta$ shown in the…

Explanation:

Step1: Identify triangle sides relative to angle $\theta$

Opposite side ($O$) = $9$, Adjacent side ($A$) = $12$.

Step2: Calculate the hypotenuse ($H$) using Pythagorean theorem

$$H = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$$

Step3: Calculate sine and cosecant values

$$\sin(\theta) = \frac{O}{H} = \frac{9}{15} = \frac{3}{5}, \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{3}$$

Step4: Calculate cosine and secant values

$$\cos(\theta) = \frac{A}{H} = \frac{12}{15} = \frac{4}{5}, \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{5}{4}$$

Step5: Calculate tangent and cotangent values

$$\tan(\theta) = \frac{O}{A} = \frac{9}{12} = \frac{3}{4}, \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{4}{3}$$

Answer:

$\sin(\theta) = \frac{3}{5}$ $\cos(\theta) = \frac{4}{5}$ $\tan(\theta) = \frac{3}{4}$ $\csc(\theta) = \frac{5}{3}$ $\sec(\theta) = \frac{5}{4}$ $\cot(\theta) = \frac{4}{3}$