The sine rule connects each side to its opposite angle: asin A= bsin B= csin C. It follows by expressing the same triangle area as 12bcsin A, 12acsin B, and 12absin C. Use it when you know a complete side–opposite-angle pair and one member of another pair. If two angles are known, find the third using the angle sum first. The rule is valid for right and non-right triangles, although right-triangle ratios are often simpler in a right triangle. Example In ABC, A=40, B=65, and b=10 cm. Then C=75, and asin40 = 10sin65 a= 10sin40 sin65 7.09 cm. The smaller side is opposite the smaller angle: 7.09<10 agrees with 40 <65. Keep the unrounded calculator result for any later calculation.

The sine rule connects each side to its opposite angle: \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\). It follows by expressing the same triangle area as \(\dfrac12bc\sin A\), \(\dfrac12ac\sin B\), and \(\dfrac12ab\sin C\).

Use it when you know a complete side–opposite-angle pair and one member of another pair. If two angles are known, find the third using the angle sum first. The rule is valid for right and non-right triangles, although right-triangle ratios are often simpler in a right triangle.

In \(\triangle ABC\), \(A=40^\circ\), \(B=65^\circ\), and \(b=10\text{ cm}\). Then \(C=75^\circ\), and \(\begin{aligned}\dfrac{a}{\sin40^\circ}&=\dfrac{10}{\sin65^\circ}\\a&=\dfrac{10\sin40^\circ}{\sin65^\circ}\\&\approx7.09\text{ cm}.\end{aligned}\)

The smaller side is opposite the smaller angle: \(7.09<10\) agrees with \(40^\circ<65^\circ\). Keep the unrounded calculator result for any later calculation.