Law of Sines

The law of sines is an essential tool in trying to solve any problem involving triangles. For example, if we are given a triangle and we know all three angles and one side, then we can use the law of sines to find the other two sides. Even more, as the following theorem shows, the law of sines is really three relationships in one statement.
Basically, the law of sines states that in any triangle, the ratio of the sine of an angle to the side opposite that angle is equal to the ratio of the sine of another angle to the side opposite that angle. The laws of sines is usually for solving triangles SSA (two sides and an angle opposite one of them) and AAS (or ASA) (two angles and any side).

Proposition (Law of Sines) Let and be the angles of a triangle. If is the length of the side opposite angle is the length of the side opposite angle and is the length of the side opposite angle then

Example (Law of Sines) When the angle of elevation of the sun is a telephone pole that is tilted at an angle of directly away from the sum casts a shadow of 21 feet long on  level ground. Approximate the length of the pole.

    Solution. The angle of elevation of the telephone is and we are given the angle opposite the telephone pole as Therefore, the angle opposite the shadow is   Let be the length of the telephone pole. Using the law of sines, we find


Since, the length of the telephone pole is approximately 33 feet.

Example (Law of Sines) In a certain isosceles triangle, the angle between the two equal sides is and the side opposite that angle is cm. Find the lengths of the two equal sides.

Solution. Let be the length of each of the two equal sides. The angles opposite the two equal sides are because Using the law of sines we have,

Since, the length of each of the two equal sides is

Example (Law of Sines) A point on level ground is 3.0 kilometers due north of a point A runner proceeds in the direction from to a point and then from to in the direction Approximate the distance run.

Solution. Let be the distance from to and let be the distance between to The angle opposite of the 3 miles is The law of sines yields

Since, the distance between Q to is 1.8 kilometers. The remaining angle of the triangle is So

Since, the distance between to is 4 kilometers. Therefore, the total distanced traveled is approximately

Cite this as:
Law Of Sines
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/law-of-sines.html