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Published byMacie Bales Modified over 2 years ago

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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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