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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.

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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

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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

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

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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

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The Law of Sines EXIT BACKNEXTTOPICS

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

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Create an altitude, h. EXIT BACKNEXTTOPICS

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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

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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

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

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General Strategies for Using the Law of Sines EXIT BACKNEXTTOPICS

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

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

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

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

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So that EXIT BACKNEXTTOPICS

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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

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By the law of sines, EXIT BACKNEXTTOPICS

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Thus, EXIT BACKNEXTTOPICS

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

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

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By the law of sines, EXIT BACKNEXTTOPICS

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

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SSA – Two Solutions EXIT BACKNEXTTOPICS

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By the law of sines, EXIT BACKNEXTTOPICS

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So that, EXIT BACKNEXTTOPICS

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

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Case 1 EXIT BACKNEXTTOPICS

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Case 2 EXIT BACKNEXTTOPICS

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Finally our two solutions: EXIT BACKNEXTTOPICS

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SSA – One Solution EXIT BACKNEXTTOPICS

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By the law of sines, EXIT BACKNEXTTOPICS

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Note– Only one is legitimate! EXIT BACKNEXTTOPICS

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Thus we have only one triangle. EXIT BACKNEXTTOPICS

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By the law of sines, EXIT BACKNEXTTOPICS

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Finally, we have: EXIT BACKNEXTTOPICS

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End of Law of Sines Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA 91754 Phone: (323) 265-8784 Email Us At: menteprog@hotmail.com Our Website: http://www.matematicamente.org EXIT BACKNEXTTOPICS

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In section 9.2 we mentioned that by the SAS condition for congruence, a triangle is uniquely determined if the lengths of two sides and the measure of.

In section 9.2 we mentioned that by the SAS condition for congruence, a triangle is uniquely determined if the lengths of two sides and the measure of.

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