1. **State the problem:** We have a right triangle with legs 33 (opposite side to angle $\theta$) and 44 (adjacent side to angle $\theta$), and hypotenuse 55. We need to find the sine, cosine, and tangent proportions with respect to $\theta$. 2. **Recall the definitions:** - Sine of angle $\theta$ is $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - Cosine of angle $\theta$ is $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ - Tangent of angle $\theta$ is $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ 3. **Calculate sine:** $$\sin(\theta) = \frac{33}{55}$$ Simplify the fraction: $$\sin(\theta) = \frac{\cancel{11} \times 3}{\cancel{11} \times 5} = \frac{3}{5}$$ 4. **Calculate cosine:** $$\cos(\theta) = \frac{44}{55}$$ Simplify the fraction: $$\cos(\theta) = \frac{\cancel{11} \times 4}{\cancel{11} \times 5} = \frac{4}{5}$$ 5. **Calculate tangent:** $$\tan(\theta) = \frac{33}{44}$$ Simplify the fraction: $$\tan(\theta) = \frac{\cancel{11} \times 3}{\cancel{11} \times 4} = \frac{3}{4}$$ **Final answers:** - $\sin(\theta) = \frac{3}{5}$ - $\cos(\theta) = \frac{4}{5}$ - $\tan(\theta) = \frac{3}{4}$