Key takeaway: The law of sines says that in any triangle, dividing a side by the sine of the angle across from it gives the same number for all three sides, which solves triangles without a right angle.
Grade 10 – GeometryG-SRT.D.10Trigonometry
Definition of Law of Sines
The law of sines carries right-triangle trigonometry over to every triangle. Pick any side, divide it by the sine of the corner facing it, and you get a number; the other two sides give exactly the same number.
The rule is most useful when two corners and one side are known, because the missing corner follows from 180 and the ratio can then be set up. It also handles a side-side-angle situation, though that case can hide two different triangles. The longest side always faces the widest corner, which is a handy check.
Formula and notation
In symbols, the law of sines for a triangle with sides \(p\), \(q\), \(r\) facing corners \(P\), \(Q\), \(R\) reads:
\[ \frac{p}{\sin P} = \frac{q}{\sin Q} = \frac{r}{\sin R} \]
Pick the two fractions that contain your unknown and one complete pair, then solve.
Key facts
- Always pair a side with the corner directly across from it.
- Two angles plus any one side are enough to start.
- With two sides and a non-enclosed angle, check for two possible triangles.
- The widest corner faces the longest side, so use that to test the answer.
Law of Sines example
A surveyor stands first at \(P\) and then at \(Q\), two spots on one rim of a ravine, and sights a boulder \(R\) on the far side. The boulder’s distance from \(Q\) is \(p = 23\) m, and the corners measure \(P = 37^{\circ}\) and \(Q = 82^{\circ}\). How far is the boulder from \(P\)? Use \(\sin 37^{\circ} \approx 0.6018\) and \(\sin 82^{\circ} \approx 0.9903\).
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The third corner is \(R = 180 – 37 – 82 = 61^{\circ}\).
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The distance from \(P\) is side \(q\), facing \(Q\). Set \(\frac{q}{\sin 82^{\circ}} = \frac{23}{\sin 37^{\circ}}\).
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Solve: \(q = \frac{23 \times 0.9903}{0.6018} \approx 37.8\) m.
The boulder is about 37.8 m from \(P\), and the corner at \(R\) is \(61^{\circ}\).
Common mistake
Watch out
Luz writes \(\frac{q}{\sin 37^{\circ}} = \frac{23}{\sin 82^{\circ}}\) and gets \(q \approx 14.1\) m.
Why it is wrong
She crossed the pairs when applying the law of sines. Side \(p = 23\) faces \(P = 37^{\circ}\), and side \(q\) faces \(Q = 82^{\circ}\). Her answer also fails the size check: the wider corner \(82^{\circ}\) must face the longer side, so \(q\) must exceed 23.
Related terms and lessons
- Geometry · Law of Sines, Law of Cosines and Triangle AreaLessonWorksheetQuiz
- Precalculus · Laws of Sines and Cosines and Triangle AreaLessonWorksheetQuiz
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