Law of Sines
In short: The law of sines declares that every side of a triangle divided by the sine of the corner facing it produces one shared number, a key for solving slanted triangles.
GeometryTrigonometryG-SRT.D.10
What is Law of Sines?
The law of sines ties edges to corners in any triangle, square-cornered or not. Name the edges \(a, b, c\) and the corners facing them \(A, B, C\); then \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
Reach for it when you hold two corners plus one edge, or two edges plus a corner that faces one of them. That second situation can be ambiguous: the data may fit zero, one or two triangles, so test whether the supplement \(180^{\circ} – B\) also works.
How to spot it
- A slanted triangle gives one edge together with the corner facing it.
- Two corners are known, so the third follows from \(180^{\circ}\), and one edge is known.
- A surveying, sailing or navigation story gives two bearings and one distance.
- Two fractions are set equal, each with an edge on top and the sine of the facing corner below.
The method
- Label each side with the lowercase letter of the angle it faces.
- Find any missing angle with the angle sum of \(180^{\circ}\).
- Choose two ratios that include the one complete pair, and set them equal.
- Cross-multiply and solve; in the SSA case, check whether \(180^{\circ} – B\) gives a second triangle.
Law of Sines: solved example
Problem
In triangle \(ABC\), \(A = 40^{\circ}\), \(B = 65^{\circ}\) and side \(a = 12\). Find \(b\) and \(c\). Use \(\sin 40^{\circ} \approx 0.6428\), \(\sin 65^{\circ} \approx 0.9063\), \(\sin 75^{\circ} \approx 0.9659\).
Solution
The third angle is \(C = 180 – 40 – 65 = 75^{\circ}\). The known pair is \(a\) and \(A\), giving the common ratio \(\frac{12}{0.6428} \approx 18.67\).
Then \(b = 18.67 \times 0.9063 \approx 16.9\) and \(c = 18.67 \times 0.9659 \approx 18.0\). The missing sides are \(b \approx 16.9\) and \(c \approx 18.0\).
Try it
Question 1
In a triangle, \(a = 14\), \(b = 10\) and \(A = 70^{\circ}\). Find angle \(B\). Use \(\sin 70^{\circ} \approx 0.9397\).
Show the answer
\(\sin B = \frac{10 \times 0.9397}{14} \approx 0.6712\), so \(B \approx 42.2^{\circ}\). The other candidate, \(137.8^{\circ}\), would make the angle sum exceed \(180^{\circ}\) with \(70^{\circ}\), so it is rejected.
Question 2
A hiker knows two trail lengths and the turn between them. Is the law of sines a good first step?
Show the answer
No. No edge faces a known corner yet, so start with the law of cosines to get the third edge.
Law of Sines vs. Law of Cosines
This law needs a facing pair: an edge together with the corner across from it. Its cousin handles the cases without such a pair, namely two edges around a corner, or three edges and no corner. Inventory the pieces you hold, then pick the law that fits them.
Keep going
Lessons
Worksheets and quizzes
Related words
Whole dictionary: A–Z · Geometry · Trigonometry