1. Statement of the problem. Find the length of side $a$ in a right triangle where angle $A=30^\circ$ and the hypotenuse is 10. 2. Formula and rules. For right triangles use the trigonometric ratio $$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$$ Important rules: the angle given is the reference angle and the opposite side is the one across from that angle. 3. Setup equation. Identify side $a$ as the side opposite angle $A$, so write $$\sin(A)=\frac{a}{10}$$ 4. Solve step-by-step. Start with $$\sin(30^\circ)=\frac{a}{10}$$ Multiply both sides by 10 to isolate $a$. $$10\sin(30^\circ)=10\cdot\frac{a}{10}$$ Show cancellation when simplifying the right side: $$10\sin(30^\circ)=\frac{\cancel{10}a}{\cancel{10}}$$ Simplify to get $$a=10\sin(30^\circ)$$ Evaluate $\sin(30^\circ)=\frac{1}{2}$. So $$a=10\cdot\frac{1}{2}=\frac{10}{2}$$ Show cancellation when simplifying the fraction: $$\frac{10}{2}=\frac{\cancel{10}}{\cancel{2}}=5$$ Round to the nearest whole number if necessary: $a=5$. Final answer: $a=5$.