Teacher Notes    

CONCEPT 1Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

The Law of Sines and Cosines are used in many different areas because of their wide reaching application to any and all triangles.  Let us look at a few ‘real world’ problems that can be solved by these laws.


Type #1 – Surveying Problem


Type #2 – Navigation (Bearing)

Bearing can be done a few different ways but no matter which way it is presented it seems to be confusing for students. Often the bearing is not the angle that is used in the problem, usually the supplement or complement is used. This can again create some confusion for the students.



Navigation works of a bearing system – one of the ways bearings are determine is to reference them from the North/South line.  Here are a few examples.



The direction and angle is referenced from either the north or the south.

Leaving from a harbor two storage ships leave, Ship #1 leaves on a bearing of 75° NW and Ship #2 leaves on a bearing of 30° NE.  If Ship #1 is travelling at 25 mph and Ship #2 is travelling at 30 mph, after 3 hours how are apart are they (to the nearest mile)?



Type #3 – Vectors


Two forces of 692 newtons and 423 newtons act at a  point.  The resultant force is 786 newtons.  Find the angle between the forces.


The resultant vector is the sum of the two vectors. A tip to tail technique is used when finding the resultant. These vectors will create a parallelogram and then the parallelogram properties help us solve for values.



Angle Θ represents the angle between the two forces but the data gives us the resultant force.  To diagram the resultant force we put our vectors tip to tail and the resultant force or vector is from the original location to the end of the second vector.  This means that when we solve for angle ß we will subtract it from 180 to get angle Θ.  Angle ß and angle Θ are consecutive interior angles of the parallelogram so angle Θ = 180° - ß.


There are many types of problems that can be used for the Law of Sines and the Law of Cosines.