Solve $A=30^{\circ },B=45^{\circ },b=10$

Step-by-step solution

1. Step 1 · Law of sines
   $$\frac{a}{\sin 30^{\circ }}=\frac{10}{\sin 45^{\circ }}$$

   Each side is paired with its opposite angle; two angles and a side determine the triangle.

2. Step 2 · Use exact sine values
   $$a=10\cdot \frac{1/2}{\sqrt{2}/2}=5\sqrt{2}$$

   Keep radicals until the final rounding to avoid accumulated error.

3. Step 3 · Check the size
   $$a\approx 7.07<10$$

   The side opposite 30° must be shorter than the side opposite 45°. Both use the same length unit.