Law of Cosines

The of cosines in especially useful when the law of sines can not be applied: for example, in the cases of SAS (two sides and the angle between them) and SSS (all three sides). Basically the law of cosines says that the square of the length of any side of a triangle equals the sum of the squares of the lengths of the other two sides minus the product of the lengths of the other two sides and the cosine of the angle between them.

Proposition (Law of Cosines) If and are lengths of any two sides of a triangle, is the angle between those sides, and is the length of the third side opposite of then

Example (Law of Cosines) In professional baseball the diamond is a square on each side. The pitcher's plate is 60.5 feet from home plate along a line from home plate to second base. If the pitcher throws a ball to the first baseman to pick off a runner on first base, how far must he throw the ball and at what angle, relative to the first base line (the line from home plate to first base), should the first baseman be prepared for the ball to come from the pitcher?

Solution. Notice that the line from home plate tot he pitcher's plate makes a with the first base line (call this angle ). Let be the distance from the pitcher's mound to first base. Let feet be the distance from the pitcher's mound to home plate, and let feet be the length from home plate to first base. Then

Therefore, the distance from the pitcher's mound to fist base is 63.7 feet. Then from the law of sines we find

and so Thus Therefore, the angle (relative to the first baseline) is

Example (Law of Cosines) NASA communicates with satellites using radio waves in straight lines at the speed of light. NASA scientists must send a command to the Pioneer Venus orbiter when Venus is ahead of the Earth in its orbit. How far will the radio waves have to travel? How many minutes will it take this message to reach Pioneer? At what angle relative to the earth-sun line must they aim the radio waves?

Solution. Let and we are looking for the distance travelled by the radio waves from the Earth to the satellite. If represents the length from Venus to the sun then we are also looking for angle which is the angle relative to the earth-sun line must they will aim the radio waves. Knowing that the distance from Venus to the sun is and the distance form the Earth to the sun is we can use the law of cosines to find as follows,

Therefore the distance travelled by the radio waves from the Earth to the satellite is The time it will take for the radio signal is (since is constant) and so we determine,

Therefore, the time it will take for the radio waves to reach the satellite is 12.8 minutes. To find the direction in which they must aim the radio waves we will use the law of sines,

Since we know that and so the angle relative to the earth-sun line they must aim the radio waves is

Example (Law of Cosines) A vertical pole 40 feet tall stands on a hillside that makes an angle of with the horizontal. Appromixmate the minimal length of cable that will reach from the top of the pole to a point 72 feet downhill from the base of the pole.

Solution. Let be the length of the cable and notice that the angle between the pole and the 72 feet length is Using the law of cosines we determine,

Therefore, the minimal length of cable that will reach from the top of the pole to a point 72 feet downhill from the base of the pole is feet.

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