Law of Sines Solving Oblique Triangles Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College EXIT TOPICSBACKNEXT Click one of the buttons below or press the enter key © 2002 East Los Angeles College. All rights reserved.

Topics Oblique Triangle Definitions The Law of Sines General Strategies for Using the Law of Sines ASA SAA The Ambiguous Case SSA Click on the topic that you wish to view... EXIT BACKNEXTTOPICS

Trigonometry can help us solve non- right triangles as well. Non-right triangles are know as oblique triangles. There are two categories of oblique trianglesacute and obtuse. EXIT BACKNEXTTOPICS

In an acute triangle, each of the angles is less than 90º. Acute Triangles EXIT BACKNEXTTOPICS

Obtuse Triangles In an obtuse triangle, one of the angles is obtuse (between 90º and 180º). Can there be two obtuse angles in a triangle? EXIT BACKNEXTTOPICS

The Law of Sines EXIT BACKNEXTTOPICS

Consider the first category, an acute triangle (,, are acute). EXIT BACKNEXTTOPICS

Create an altitude, h. EXIT BACKNEXTTOPICS

Lets create another altitude h. EXIT BACKNEXTTOPICS

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Putting these together, we get This is known as the Law of Sines. EXIT BACKNEXTTOPICS

The Law of Sines is used when we know any two angles and one side or when we know two sides and an angle opposite one of those sides. EXIT BACKNEXTTOPICS

Fact The law of sines also works for oblique triangles that contain an obtuse angle (angle between 90º and 180º). is obtuse EXIT BACKNEXTTOPICS

General Strategies for Using the Law of Sines EXIT BACKNEXTTOPICS

One side and two angles are known. ASA or SAA EXIT BACKNEXTTOPICS

ASA From the model, we need to determine a, b, and using the law of sines. EXIT BACKNEXTTOPICS

First off, 42º + 61º + = 180º so that = 77º. (Knowledge of two angles yields the third!) EXIT BACKNEXTTOPICS

Now by the law of sines, we have the following relationships: EXIT BACKNEXTTOPICS

So that EXIT BACKNEXTTOPICS

SAA From the model, we need to determine a, b, and using the law of sines. Note: + 110º + 40º = 180º so that = 30º a b EXIT BACKNEXTTOPICS

By the law of sines, EXIT BACKNEXTTOPICS

Thus, EXIT BACKNEXTTOPICS

The Ambiguous Case – SSA In this case, you may have information that results in one triangle, two triangles, or no triangles. EXIT BACKNEXTTOPICS

SSA – No Solution Two sides and an angle opposite one of the sides. EXIT BACKNEXTTOPICS

By the law of sines, EXIT BACKNEXTTOPICS

Thus, Therefore, there is no value for that exists! No Solution! EXIT BACKNEXTTOPICS

SSA – Two Solutions EXIT BACKNEXTTOPICS

By the law of sines, EXIT BACKNEXTTOPICS

So that, EXIT BACKNEXTTOPICS

Case 1 Case 2 Both triangles are valid! Therefore, we have two solutions. EXIT BACKNEXTTOPICS

Case 1 EXIT BACKNEXTTOPICS

Case 2 EXIT BACKNEXTTOPICS

Finally our two solutions: EXIT BACKNEXTTOPICS

SSA – One Solution EXIT BACKNEXTTOPICS

By the law of sines, EXIT BACKNEXTTOPICS

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Note– Only one is legitimate! EXIT BACKNEXTTOPICS

Thus we have only one triangle. EXIT BACKNEXTTOPICS

By the law of sines, EXIT BACKNEXTTOPICS

Finally, we have: EXIT BACKNEXTTOPICS

End of Law of Sines Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA Phone: (323) Us At: Our Website: EXIT BACKNEXTTOPICS