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There is no doubt that the 3 PTRs are extremely useful when solving problems modeled on a right triangle. Unfortunately, the world does not consist only of right triangles…

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As a matter of fact, right triangles end up being more of a rarity than commonplace. Does that mean when we come across a situation that can only be modeled with a non-right triangle that we abandon our pursuit?….

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No Way!!!! There exists 2 Laws of Trigonometry that allow one to solve problems that involve non-right Triangles:

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A triangle is uniquely determined by two angles and a particular side A C B c b a O1O1 O2O2

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If a corresponding angle and side are known, they form an “opposing pair” A C B c b a O1O1 O2O2

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The Sine Law can be used to determine an unknown side or angle given an “opposing pair” A C B c b a O1O1 O2O2

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Find the length of b A C B c b 5 30 o 65 o

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Construct CN with height h A C B c b 5 30 o 65 o N h

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By the right triangle SIN ratio A C B c b 5 30 o 65 o N h Sin 30 o = h b Sin 65 o = h 5 h 30 o 65 o

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Solve both equations for h X b Sin 30 o = h b Sin 65 o = h 5 X 5 bSin30 o = h h = 5Sin65 o Because the equations are equal bSin30 o = 5Sin65 o

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b = 5Sin65 o Sin30 o b = 9.1 Consider the general case:

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A C c b a N h B Sin A = h b Sin B = h a bSinA = aSinB

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a a bSinA = SinB a b ab b SinA = SinB a

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Extend this to all 3 sides of a triangle, and the Sine Law is generated! b SinA = SinB a c = SinC

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Find the length of a a A C c 24 73 o 57 o N a Sin73 o =24 Sin57 o a = 27.4

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5.9 O 10.3 O 2.9 km Find h h

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5.9 O 10.3 O 2.9 km Find h 1. Find O O O = 180 O – 5.9 O – 10.3 O = 163.8 O

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5.9 O 10.3 O 2.9 km 163.8 O Find X X X SIN 10.3 O 2.9 SIN163.8 O = X = 1.86km

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5.9 O 10.3 O 2.9 km Find h h 1.86 km SIN 5.9 O = h 1.86 km h = 191.2 m

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The Ambiguous Case

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Find A 11 9 48 o A 11 SinA = 9 Sin48 o A=65.3 o Does that make sense? No Way!!!

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Side 9 can also be drawn as: 11 9 48 o A Could A be 65 o in this case?

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This type of discrepancy is called the “Ambiguous Case” Be sure to check the diagram to see which answer fits: O, or 180 o - O

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What you’ll learn Use the Law of Sines to solve oblique triangles. Use the Law of Sines to solve, if possible, the triangle or triangles in the ambiguous.

What you’ll learn Use the Law of Sines to solve oblique triangles. Use the Law of Sines to solve, if possible, the triangle or triangles in the ambiguous.

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