Trigonometry/Law of Sines

< Trigonometry

Law-of-sines1.svg

For any triangle with vertices corresponding angles and corresponding opposite side lengths , the Law of Sines states that

Each of these expressions is also equal to the diameter of the triangle's circumcircle (the circle that passes through the points ). The law can also be written in terms of the reciprocals:

ProofEdit

 

Dropping a perpendicular   from vertex   to intersect   (or   extended) at   splits this triangle into two right-angled triangles   and   . We can calculate the length   of the altitude   in two different ways:

  • Using the triangle AOC gives
  ;
  • and using the triangle BOC gives
  .
  • Eliminate   from these two equations:
  .
  • Rearrange to obtain
 

By using the other two perpendiculars the full law of sines can be proved. QED.

ApplicationEdit

This formula can be used to find the other two sides of a triangle when one side and the three angles are known. (If two angles are known, the third is easily found since the sum of the angles is   .) See Solving Triangles Given ASA. It can also be used to find an angle when two sides and the angle opposite one side are known.

Area of a triangleEdit

The area of a triangle may be found in various ways. If all three sides are known, use Heron's theorem.

If two sides and the included angle are known, consider the second diagram above. Let the sides   and   , and the angle between them   be known. The terms /alpha and /gamma are variables represented by Greek alphabet letters, and these are commonly used interchangeably in trigonometry just like English variables x, y, z, a, b, c, etc. From triangle   , the altitude   is   so the area is   .

If two angles and the included side are known, again consider the second diagram above. Let the side   and the angles   and   be known. Let   . Then

 

Thus

  .