Prove the Laws of Sines and Cosines and use them to solve problems.
Use the Law of Sines to find a missing side from two angles and one opposite side
Problem
In the triangle shown, \(A~=~30°\), \(B~=~45°\), and \(a~=~10\). Use the Law of Sines to find \(b\), giving an exact value and a decimal approximation.
Build the Law of Sines proportion from opposite pairs, not from where labels happen to sit in the drawing. Anchor the proportion with the fully known side-angle pair and match the requested side with its opposite angle. Solve before substituting special-angle values, preserving an exact form until the final decimal check.
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