1. **Problem Statement:** Determine the length of side $c$ in the right triangle with $\angle C = 90^\circ$, $BC = 5$ ft, $\angle A = 30^\circ$, and side $BA = c$. 2. **Using the Sine Ratio:** The sine ratio formula is: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ Here, $\sin A = \frac{\text{opposite side to } A}{\text{hypotenuse}} = \frac{BC}{BA} = \frac{5}{c}$. Rearranging to solve for $c$: $$c = \frac{5}{\sin 30^\circ}$$ Since $\sin 30^\circ = 0.5$: $$c = \frac{5}{0.5} = 10$$ 3. **Using the Sine Law:** The sine law states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ In triangle $ABC$, $C = 90^\circ$, so $\sin C = 1$. We know side $a = BC = 5$, angle $A = 30^\circ$, and side $c = BA$ is unknown. Using the sine law: $$\frac{c}{\sin C} = \frac{a}{\sin A}$$ Substitute values: $$\frac{c}{1} = \frac{5}{\sin 30^\circ}$$ $$c = \frac{5}{0.5} = 10$$ 4. **Why do both methods have the same result?** Because $\sin 90^\circ = 1$, the sine law simplifies to the sine ratio in right triangles, making both methods equivalent. 5. **Is the sine law true for right triangles?** Yes, the sine law holds true for right triangles as a special case where one angle is $90^\circ$. --- 6. **Jace's Mosaic Tile Problem:** Given a triangle with sides $AB = 9.1$ cm, $BC = 11.4$ cm, and $\angle A = 64^\circ$, find $\angle C$. Use the sine law: $$\frac{AB}{\sin C} = \frac{BC}{\sin A}$$ Rearranged: $$\sin C = \frac{AB \cdot \sin A}{BC} = \frac{9.1 \times \sin 64^\circ}{11.4}$$ Calculate $\sin 64^\circ \approx 0.8988$: $$\sin C = \frac{9.1 \times 0.8988}{11.4} = \frac{8.179}{11.4} \approx 0.7173$$ Find $\angle C$: $$C = \sin^{-1}(0.7173) \approx 45.8^\circ$$ --- 7. **Amy's City Planning Problem:** Amy wants to find the length of the new road $EO$ in triangle $REO$ with sides $RE$ (Oak St.) and $RO$ (Elm St.) and angle $E$. Using the sine law: $$\frac{EO}{\sin R} = \frac{RE}{\sin O} = \frac{RO}{\sin E}$$ To calculate $EO$, Amy needs: - The lengths of $RE$ and $RO$ - The measure of angle $E$ - The measure of angle $R$ or $O$ to apply the sine law properly **Summary:** Amy can use the sine law to find $EO$ if she knows two sides and an angle not between them or two angles and one side.