**THE LAWS OF SINES AND COSINES
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Solve oblique triangles using the Law of Sines.

2. Solve oblique triangles using the Law of Cosines.

In a previous lesson, we learned how to use the trigonometric ratios and the

Pythagorean Theoremto find missing sides and angles in a right triangle. We called this process "solving a triangle." Now, we will learn how to do the same for oblique triangles, that is, triangles that do NOT contain a right angle.Some oblique triangles can be solved using the

Law of Sinesfor others we must use theLaw of Cosines. These laws can also be used with right triangles.

The Law of Sines- see #1 through 3 in the "Examples" documentTo use the

Law of Sines, we must be given at least one angle and its opposite side. You do not have to memorize theLaw of Sines, however, you must know how to work with it when given.Given a triangle with angles

,A, andB, and opposite sides labeledC,a, andb, as shown in the picture below, thencor

The proof for this law is provided separately. If you are interested, it can be found under the link "Point of Interest 4" in the

Learning Materials#15 in theMyOpenMathcourse.

Please note that the Law of Sines sometimes produces two different sets of solutions!

The Law of Cosines- see #4 and 5 in the "Examples" documentWhen the

Law of Sinescannot be applied, we must use theLaw of Cosines. You do not have to memorize theLaw of Cosines, however, you must know how to work with it when given.

NOTE: Once we start using this law, we continue to do so for all calculations even if we could switch to the Law of Sines eventually.Given a triangle with angles

,A, andB, and opposite sides labeledC,a, andb, as shown in the picture below, thenc

Likewise, we can write the

Law of Cosinesfor the other two sides as follows:

The proof for this law is provided separately. If you are interested, it can be found under the link "Point of Interest 5" in the

Learning Materials#15 in theMyOpenMathcourse.