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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 triangles—acute and obtuse. TOPICS. BACK. NEXT. EXIT. Acute Triangles. In an acute triangle, each of the angles is less than 90 º. TOPICS.
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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 triangles—acute and obtuse. TOPICS BACK NEXT EXIT
Acute Triangles In an acute triangle, each of the angles is less than 90º. TOPICS BACK NEXT EXIT
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? TOPICS BACK NEXT EXIT
The Law of Sines TOPICS BACK NEXT EXIT
Consider the first category, an acute triangle (, , are acute). TOPICS BACK NEXT EXIT
Create an altitude, h. * * * * TOPICS BACK NEXT EXIT
Let’s create another altitude h’. TOPICS BACK NEXT EXIT
* * * * TOPICS BACK NEXT EXIT
Putting these together, we get This is known as the Law of Sines. TOPICS BACK NEXT EXIT
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. TOPICS BACK NEXT EXIT
Fact The law of sines also works for oblique triangles that contain an obtuse angle (angle between 90º and 180º). is obtuse TOPICS BACK NEXT EXIT
General Strategies for Usingthe Law of Sines TOPICS BACK NEXT EXIT
One side and two angles are known. ASA or SAA TOPICS BACK NEXT EXIT
ASA From the model, we need to determine a, b, and using the law of sines. TOPICS BACK NEXT EXIT
First off, 42º + 61º + = 180º so that = 77º. (Knowledge of two angles yields the third!) TOPICS BACK NEXT EXIT
Now by the law of sines, we have the following relationships: TOPICS BACK NEXT EXIT
So that ● ● ● ● TOPICS BACK NEXT EXIT
SAA From the model, we need to determine a, b, and using the law of sines.Note: + 110º + 40º = 180º so that = 30º b a TOPICS BACK NEXT EXIT
By the law of sines, TOPICS BACK NEXT EXIT
Thus, ● ● ● ● TOPICS BACK NEXT EXIT
The Ambiguous Case – SSA In this case, you may have information that results in one triangle, two triangles, or no triangles. TOPICS BACK NEXT EXIT
SSA – No Solution Two sides and an angle opposite one of the sides. TOPICS BACK NEXT EXIT
By the law of sines, TOPICS BACK NEXT EXIT
Thus, Therefore, there is no value for that exists! No Solution! TOPICS BACK NEXT EXIT
SSA – Two Solutions TOPICS BACK NEXT EXIT
By the law of sines, TOPICS BACK NEXT EXIT
So that, TOPICS BACK NEXT EXIT
Case 1Case 2 Both triangles are valid! Therefore, we have two solutions. TOPICS BACK NEXT EXIT
Case 1 * * TOPICS BACK NEXT EXIT
Case 2 * * TOPICS BACK NEXT EXIT
Finally our two solutions: TOPICS BACK NEXT EXIT
SSA – One Solution TOPICS BACK NEXT EXIT
By the law of sines, TOPICS BACK NEXT EXIT
* * TOPICS BACK NEXT EXIT
Note– Only one is legitimate! TOPICS BACK NEXT EXIT
Thus we have only one triangle. TOPICS BACK NEXT EXIT
By the law of sines, * * TOPICS BACK NEXT EXIT
Finally, we have: TOPICS BACK NEXT EXIT
End of Law of SinesHomework – Pg 484 1-11 odd, 13-15, 19, 23, 27, 28, 37-39, 45 TOPICS BACK NEXT EXIT