1. **Problem:** Find the six trigonometric function values of the specified angle given the triangle with angle 30° between side 18 (vertical) and side 24 (horizontal), right angle adjacent to side 24, angle at right angle labeled $\theta$. 2. **Step 1: Identify sides relative to $\theta$** - Opposite side to $\theta$ is 18 (vertical side). - Adjacent side to $\theta$ is 24 (horizontal side). - Hypotenuse $c$ is found by Pythagoras theorem: $$c = \sqrt{18^2 + 24^2} = \sqrt{324 + 576} = \sqrt{900} = 30$$ 3. **Step 2: Write formulas for six trig functions:** - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ - $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$ - $\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$ - $\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}$ 4. **Step 3: Calculate each value:** - $\sin \theta = \frac{18}{30} = \frac{3}{5} = 0.6$ - $\cos \theta = \frac{24}{30} = \frac{4}{5} = 0.8$ - $\tan \theta = \frac{18}{24} = \frac{3}{4} = 0.75$ - $\csc \theta = \frac{30}{18} = \frac{5}{3} \approx 1.6667$ - $\sec \theta = \frac{30}{24} = \frac{5}{4} = 1.25$ - $\cot \theta = \frac{24}{18} = \frac{4}{3} \approx 1.3333$ 5. **Final answer:** \[ \sin \theta = 0.6, \quad \cos \theta = 0.8, \quad \tan \theta = 0.75, \quad \csc \theta = 1.6667, \quad \sec \theta = 1.25, \quad \cot \theta = 1.3333 \]